The Radon Transform

Second Edition

Progress in Mathematics 5

Sigurdur Helgason

The Radon Transform

Progress in Mathematics Volume5

Series Editors Hyman Bass Joseph Oesterle Alan Weinstein

Sigurdur Helgason

The Radon Transform Second Edition

Springer Science+Business Media, LLC

Sigurdur Helgason Department of Mathematics MIT Cambridge, MA 02139 USA

Library of Congress Cataloging-in-Publication Data Helgason, Sigurdur, 1927

The Radon transform I Sigurdur Helgason. -- 2nd ed. p. em. -- (Progress in mathematics ; v. 5) Includes bibliographical references and index.

ISBN 978-1-4757-1465-4 ISBN 978-1-4757-1463-0 (eBook) DOI 10.1007/978-1-4757-1463-0

1. Radon transforms. I. Title. II. Series: mathematics (Boston, Mass.) ; vol. 5.

QA649.H44 1999 515'. 723--dc21

Progress in

99-29331 CIP

AMS Subject Classifications: Primary: 22E30, 35L05, 43A85, 44A12, 53C65 Secondary: 22E46, 53C35, 92C55

Printed on acid-free paper. © 1999 Sigurdur Helgason, Second Edition Originally published by Birkhiiuser Boston in 1999

@1980 Sigurdur Helgason, First Edition

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ISBN 978-1-4757-1465-4 SPIN 19901615

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CONTENTS

Preface to the Second Edition ..................................... ix

Preface to the First Edition ....................................... xi

CHAPTER I

The Radon Transform on JR.n

1. Introduction ...................................................... 1

2. The Radon Transform of the Spaces V(JR.n) and S(JR.n). The Support Theorem ............................................ 2

3. The Inversion Formula ........................................... 15

4. The Plancherel Formula ......................................... 20

5. Radon Transform of Distributions ............................... 22

6. Integration over d-Planes. X-ray Transforms. The Range of the d-Plane Transform ............................ 28

7. Applications ..................................................... 41

a) Partial Differential Equations ............................... 41

b) X-ray Reconstruction ....................................... 46

Bibliographical Notes ............................................ 51

CHAPTER II

A Duality in Integral Geometry.

Generalized Radon Transforms and Orbital Integrals

1. Homogeneous Spaces in Duality ................................. 53

2. The Radon Transform for the Double Fibration .................. 57

3. Orbital Integrals ................................................. 62

vi Contents

4. Examples of Radon Transforms for Homogeneous Spaces in Duality .............................. 63

A. The Funk Transform ....................................... 63

B. The X-ray Transform in H 2 ••.•••.••••••••.•••••••••••••••• 66

C. The Horocycles in H 2 •.•••••••••••.•.••••.•••••.••••••••••• 67

D. The Poisson Integral as a Radon Transform ................. 71

E. The d-Plane Transform ..................................... 73

F. Grassmann Manifolds ....................................... 74

G. Half-lines in a Half-plane ................................... 75

H. Theta Series and Cusp Forms ............................... 79

Bibliographical Notes ............................................ 80

CHAPTER III

The Radon Transform on Two-Point Homogeneous Spaces

1. Spaces of Constant Curvature. Inversion and Support Theorems ................................ 83

A. The Hyperbolic Space ...................................... 85

B. The Spheres and the Elliptic Spaces ........................ 93

C. The Spherical Slice Transform ............................. 108

2. Compact Two-Point Homogeneous Spaces. Applications ........ 111

3. Noncompact Two-Point Homogeneous Spaces ................... 118

4. The X-ray Transform on a Symmetric Space .................... 119

5. Maximal Tori and Minimal Spheres in Compact Symmetric Spaces .................................. 120

Bibliographical Notes ........................................... 122

CHAPTER IV

Orbital Integrals and the Wave Operator for Isotropic Lorentz Spaces

1. Isotropic Spaces ................................................ 123

Contents vii

A. The Riemannian Case ..................................... 124

B. The General Pseudo-Riemannian Case ..................... 124

C. The Lorentzian Case ...................................... 128

2. Orbital Integrals ............................................... 128

3. Generalized Riesz Potentials .................................... 137

4. Determination of a Function from Its Integrals over Lorentzian Spheres ........................................ 140

5. Orbital Integrals and Huygens' Principle ........................ 144


CHAPTER V

Fourier Transforms and Distributions. A Rapid Course

1. The Topology of the Spaces 'D(Rn), £(Rn) and S(Rn) .......... 147

2. Distributions ................................................... 149

3. The Fourier Transform ......................................... 150

4. Differential Operators with Constant Coefficients ............... 156

5. Riesz Potentials ................................................ 160


Bibliography .•...................•...................•..•.... 71

Notational Conventions .............•.....•........•......•• 185

Subject Index ..................•......•.................... 187

PREFACE TO THE SECOND EDITION

The first edition of this book has been out of print for some time and I have decided to follow the publisher's kind suggestion to prepare a new edition. Many examples with explicit inversion formulas and range theorems have been added, and the group-theoretic viewpoint emphasized. For example, the integral geometric viewpoint of the Poisson integral for the disk leads to interesting analogies with the X-ray transform in Euclidean 3-space. To preserve the introductory flavor of the book the short and self-contained Chapter Von Schwartz' distributions has been added. Here §5 provides proofs of the needed results about the Riesz potentials while §§3-4 develop the tools from Fourier analysis following closely the account in Hormander's books (1963] and [1983]. There is some overlap with my books (1984] and [1994b] which however rely heavily on Lie group theory. The present book is much more elementary.

I am indebted to Sine Jensen for a critical reading of parts of the manuscript and to Hilgert and Schlichtkrull for concrete contributions mentioned at specific places in the text. Finally I thank Jan Wetzel and Bonnie Friedman for their patient and skillful preparation of the manuscript.

Cambridge, 1999

PREFACE TO THE FIRST EDITION

The title of this booklet refers to a topic in geometric analysis which has its origins in results of Funk [1916) and Radon [1917] determining, respectively, a symmetric function on the two-sphere 8 2 from its great circle integrals and a function on the plane R 2 from its line integrals. (See references.) Recent developments, in particular applications to partial differential equations, X-ray technology, and radio astronomy, have widened interest in the subject.

These notes consist of a revision of lectures given at MIT in the Fall of 1966, based mostly on my papers during 1959-1965 on the Radon transform and its generalizations. (The term "Radon Thansform" is adopted from John [1955].)

The viewpoint for these generalizations is as follows. The set of points on 8 2 and the set of great circles on 8 2 are both

homogeneous spaces of the orthogonal group 0(3). Similarly, the set of points in R 2 and the set of lines in R 2 are both homogeneous spaces of the group M(2) of rigid motions of R 2 • This motivates our general Radon transform definition from [1965a, 1966a] which forms the framework of Chapter II: Given two homogeneous spaces G I K and G I H of the same group G, the Radon transform u-+ u maps functions u on the first space to functions u on the second space. For ~ E G I H, u( ~) is defined as the (natural) integral of u over the set of points x E G I K which are incident to ~in the sense of Chern [1942]. The problem of inverting u-+ u is worked out in a few cases.

It happens when· G I K is a Euclidean space, and more generally when G I K is a Riemannian symmetric space, that the natural differential operators A on G I K are transferred by u -+ u into much more mangeable differential operators A on Gl H; the connection is (Au)"= Au. Then the theory of the transform u -+ u has significant applications to the study of properties of A.

On the other hand, the applications of the original Radon transform on R 2 to X-ray technology and radio astronomy are based on the fact that for an unknown density u, X-ray attenuation measurements give u directly and therefore yield u via Radon's inversion formula. More precisely, let B be a convex body, u(x) its density at the point x, and suppose a thin beam of X-rays is directed at B along a line~· Then the line integral u(~) of u along~ equals log(Ioii) where Io and I, respectively, are the intensities of the beam before hitting B and after leaving B. Thus while the function u is at first unknown, the function u is determined by the X-ray data.

xii Preface

The lecture notes indicated above have been updated a bit by the inclusion of a short account of some applications (Chapter I, §7), by adding a few corollaries (Corollaries 2.8 and 2.12, Theorem 6.3 in Chapter I, Corollaries 2.8 and 4.1 in Chapter III), and by giving indications in the bibliographical notes of some recent developments.

An effort has been made to keep the exposition rather elementary. The distribution theory and the theory of Riesz potentials, occasionally needed in Chapter I, is reviewed in some detail in §8 (now Chapter V). Apart from the general homogeneous space framework in Chapter II, the treatment is restricted to Euclidean and isotropic spaces (spaces which are "the same in all directions"). For more general symmetric spaces the treatment is postponed (except for §4 in Chapter III) to another occasion since more machinery from the theory of semisimple Lie groups is required.

I am indebted to R. Melrose and R. Seeley for helpful suggestions and to F. Gonzalez and J. Orloff for critical reading of parts of the manuscript.

Cambridge, MA 1980

The Radon Transform

CHAPTER I

THE RADON TRANSFORM ON JRn

§1 Introduction

It was proved by J. Radon in 1917 that a differentiable function on JR3 can be determined explicitly by means of its integrals over the planes in JR.3 • Let J(w,p) denote the integral off over the hyperplane (x,w) = p, w denoting a unit vector and ( , ) the inner product. Then

f(x) = - 8~2 L, (fs2

J(w, (w, x)) dMJ) ,

where L is the Laplacian on JR3 and dM; the area element on the sphere 8 2

(cf. Theorem 3.1). We now observe that the formula above has built in a remarkable du

ality: first one integrates over the set of points in a hyperplane, then one integrates over the set of hyperplanes passing through a given point. This suggests considering the transforms f -+ j, <p -+ if; defined below.

The formula has another interesting feature. For a fixed w the integrand x -+ J(w, (w, x)) is a plane wave, that is a function constant on each plane perpendicular to w. Ignoring the Laplacian the formula gives a continuous decomposition of f into plane waves. Since a plane wave amounts to a function of just one variable (along the normal to the planes) this decomposition can sometimes reduce a problem for JR.3 to a similar problem for JR. This principle has been particularly useful in the theory of partial differential equations.

The analog of the formula above for the line integrals is of importance in radiography where the objective is the description of a density function by means of certain line integrals.

In this chapter we discuss relationships between a function on JR.n and its integrals over k-dimensional planes in JR.n . The case k = n - 1 will be the one of primary interest. We shall occasionally use some facts about Fourier transforms and distributions. This material will be devdoped in sufficient detail in Chapter V so the treatment should be self-contained.

Following Schwartz [1966] we denote by £(JR.n) and V(JR.n ), respectively, the space of complex-valued coo functions (respectively coo functions of compact support) on JR.n • The space S (JR.n) of rapidly decreasing functions on JR.n is defined in connection with ( 6) below. em (JR.fl) denotes the space of m times continuously differentiable functions. We write C (JR.n) for C0 (JR.n), the space of continuous function on JR.n •

For a manifold M, em ( M) (and C ( M)) is defined similarly and we write V(M) for Cg"(M) and £(M) for C00 (M).

2 Chapter I. The Radon Transform on Rn

§2 The Radon Transform of the Spaces V(JRn) and S(JRn). The Support Theorem

Let I be a function on JRn, integrable on each hyperplane in JRn . Let pn denote the space of all hyperplanes in JRn, pn being furnished with the obvious topology. The Radon transform of I is defined as the function f on pn given by

f(e) = l l(x)dm(x),

where dm is the Euclidean measure on the hyperplane e. Along with the transformation I -+ f we consider also the dual transform <p -+ ip which to a continuous function <p on pn associates the function ip on JRn given by

where df..L is the measure on the compact set { e E pn : x E e} which is invariant under the group of rotations around x and for which the measure of the whole set is 1 (see Fig. I.l). We shall relate certain function spaces on JRn and on pn by means of the transforms I -+ j, <p -+ if;; later we obtain explicit inversion formulas.

FIGURE 1.2.

FIGURE 1.1.

Each hyperplane e E pn can be written e = { x E lRn : (x, w) = p} where ( , ) is the usual inner product, w = (w1 , ... , wn) a unit .. vector and p E lR (Fig. !.2). Note that the pairs (w,p) and (-w, -p) give the same e; the mapping (w,p) -+ e is a double COVering of sn-l X JR onto pn. Thus pn has a canonical manifold structure with respect to which this covering map is differentiable and regular. We thus identify continuous

§2 The Radon Transform . . . The Support Theorem 3

(differentiable) functions c.p on pn with continuous (differentiable) functions c.p on P with continuous (differentiable) functions c.p on gn-l x R. satisfying c.p(w,p) = c.p( -w, -p). Writing 1(w,p) instead of 1(~) and ft (with t E R.n) for the translated function x -+ f ( t + x) we have

h(w,p) = r f(x + t) dm(x) = r f(y) dm(y) J(x,w)=p }(y,w)=p+(t,w)

so

(1) h(w,p) = J(w,p + (t,w)).

Taking limits we see that if Oi = a I OXi

(2) - a1 (8d)(w,p) = Wi op (w,p).

Let L denote the Laplacian "Ei8T on R.n and let 0 denote the operator

82 c.p(w,p) -+ op2 c.p(w,p),

which is a well-defined operator on £(Pn) = C00 (Pn). It can be proved that if M(n) is the group of isometries of R.n, then L (respectively D) generates the algebra of M(n)-invariant differential operators on R.n (respectively pn).

Lemma 2.1. The transforms f -+ 1, c.p-+ (/; intertwine L and 0, i.e.,

Proof. The first relation follows from (2) by iteration. For the second we just note that for a certain constant c

(3) {/;(x) = c r c.p(w, (x,w)) dw' }sn-1

where dw is the usual measure on sn-l.

The Radon transform is closely connected with the Fourier transform

f{u) = { f(x)e-i\x,w) dx u E R.n. }JRn

In fact, if s E JR, w a unit vector,

f(sw) = roo dr r f(x)e-is(x,w) dm(x) } -oo J(x,w)=r

4 Chapter I. The Radon Transform on R.n

so

(4) f(sw) = /_: f(w,r)e-isr dr.

This means that the n-dimensional Fourier transform is the !-dimensional Fourier transform of the Radon transform. From ( 4), or directly, it follows that the Radon transform of the convolution

is the convolution

(5)

f(x) = r h(x- y)h(y) dy jR,.

f(w,p) = k h (w,p- q)h(w, q) dq.

We consider now the space S(Rn) of complex-valued rapidly decreasing functions on Rn. We recall that f E S(r) if and only if for each polynomial P and each integer m ~ 0,

(6) sup llxlm P(8t, ... , On)f(x)l < oo, :z:

lxl denoting the norm of x. We· now formulate this in a more invariant fashion.

Lemma 2.2. A function f E £(Rn) belongs to S(Rn) if and only if for each pair k, l E z+

sup 1(1 + !xl)k(Ll f)(x)l < oo. :z:ER."

This is easily proved just by using the Fourier transforms. In analogy with S(Rn) we define S(sn-1 x R) as the space of coo functions

cp on gn-1 x R which for any integers k, l ~ 0 and any differential operator D on gn-1 satisfy

(7) sup 1(1+ lrlk)dd!l(Dcp)(w,r)l < oo. wES"-l,rER. T

The space S{Pn} is then defined as the set of cp E S(sn-1 x R) satisfying cp(w,p) = cp( -w, -p).

Lemma 2.3. For each f E S(r) the Radon transform j(w,p) satisfies the following condition: For k E Z + the integral

k j(w,p)pk dp

can be written as a kth degree homogeneous polynomial in w1, ... , Wn.


Proof. This is immediate from the relation

(8) r 1(w,p)pk dp = r pk dp r f(x) dm(x) = r f(x)(x,w)k dx. A~. JR J(z,w)=p JR"

In accordance with this lemma we define the space

{ For each k E z+,JRF(w,p)pkdp }

S H (Pn) = F E S (Pn) : is a homogeneous polynomial . in w1, ... , Wn of degree k

With the notation 1J(Pn) = C~(Pn) we write

According to Schwartz [1966], p. 249, the Fourier transform f-+ J maps the space S(Jm.n) onto itself. See Ch. V, Theorem 3.1. We shall now settle the analogous question for the Radon transform.

Theorem 2.4. {The Schwartz theorem) The Radon transform f-+ 1 is a linear one-to-one mapping of S(Jm.n) onto SH(Pn).

Proof. Since

it is clear from (4) that for each fixed w the function r -+ 1(w, r) lies in S{IR). For each w0 E sn-1 a subset of (w1. ... ,wn) will serve as local coordinates on a neighborhood of w0 in sn-1. To see that 1 E S{!Rn ), it therefore suffices to verify (7) for <p = 1 on an open subset N C sn-1

where Wn is bounded away from 0 and w1, ... , Wn-1 serve as coordinates, in terms of which D is expressed. Since

(9) u1 = sw1, ... , Un-1 = SWn-1, Un = s(1- wi - · · ·- w~_1 ) 1/2

we have

a (-( )) aj ( 2 2 )-1;2 aj - f sw = s- - swi 1 - w1 - • · · - wn_1 -. awi aui 8un

It follows that if D is any differential operator on sn-l and if k, l E z+ then

(10) I dl _ I sup (1 + s2k)-d l (D f)(w, s) < oo.

wEN,sER S


We can therefore apply D under the integral sign in the inversion formula to (4),

J(w,r) = 2~ L l{sw)eisr ds

and obtain

(1+r2k) ::l (Dw(1(w,r))) = 2~ /(1+(-1)k ~:k )((is)lDw(J(sw)))eisrds.

Now (10) shows that 1 E S(Pn) so by Lemma 2.3, 1 E SH(Pn). Because of (4) and the fact that the Fourier transform is one-to-one it

only remains to prove the surjectivity in Theorem 2.4. Let cp E SH(Pn). In order to prove cp = 1 for some f E S(lRn) we put

'1/J(s,w) =I: cp(w,r)e-irsdr.

Then '1/J(s,w) = '1/J(-s,-w) and 1/J(O,w) is a homogeneous polynomial of degree 0 in w1o ••• , Wm hence constant. Thus there exists a function F on JR.n such that

F(sw) = L cp(w,r)e-irsdr.

While F is clearly smooth away from the origin we shall now prove it to be smooth at the origin too; this is where the homogeneity condition in the definition of SH(Pn) enters decisively. Consider the coordinate neighborhood N C gn-1 above and if hE C00 (lR.n - 0) let h*(w1, ... ,wn-1, s) be the function obtained from h by means of the substitution (9). Then

and

8ui Hence

8h n-1 8h* 8wj 8h* 8s --"'--+-·- (1 :::;i :::;n) 8u· - L.J 8w· 8u· 8s 8u· • j=1 J • •

1(~ UiUj) ( . ) - u· · - -- 1 < z < n 1 :::; j :::; n- 1 , s '3 s2 - - '

( . )· 8s ( 2 2 ) 1/2 = w· 1 < z < n- 1 - = 1- w1 - .. · - w 1 ' - - '8un n- ·

1 8h* (8h* 1 n-l 8h*) = -; 8wi + Wi 8s - -; ~ Wj 8wJ· (1 :::; i :::; n- 1)

J=1

(8h* 1 n-1 8h*) = (1-w2-···-w2_1)1/2 ---"'w·- .

1 n 8s s L.J J 8w. j=l J


In order to use this for h = F we write

F(sw) = /_: rp(w,r) dr + /_: rp(w,r)(e-irs -1) dr.

By assumption the first integral is independent of w. Thus using (7) we have for constant K > 0

and a similar estimate is obvious for 8F(sw)f8s. The formulas above therefore imply that all the derivatives 8F I aui are bounded in a punctured ball 0 < lui < € so F is certainly continuous at u = 0.

More generally, we prove by induction that

(11)

where the coefficients A have the form

(12)

For q = 1 this is in fact proved above. Assuming (11) for q we calculate

aq+~h

using the above formulas for 8f8ui. If Aj,k1 ••• k; (w, s) is differentiated with respect to 'Uiq+t we get a formula like (12) with q replaced by q + 1. If on the other hand the (i + j)th derivative of h* in (11) is differentiated with respect to 'Uiq+t we get a combination of terms

-1 8i+i+lh* 8i+i+1h*

s awkt ... awki+18si ' awkt ... awk,8si+1

and in both cases we get coefficients satisfying (12) with q replaced by q+ 1. This proves (11)-(12) in general. Now

roo q-1 ( . )k ro {13) F(sw)= }_

00rp(w,r);; -~r dr+ }_

00rp(w,r)eq(-irs)dr,

where

tq tq+1

eq(t) = I+ ( 1)' + ... q. q+ .


Our assumption on rp implies that the first integral in (13) is a polynomial in u1, .•. , un of degree ~ q- 1 and is therefore annihilated by the differential operator (11). If 0 ~ j ~ q, we have

f)i . !si-q~(eq(-irs))i = i(-ir)q(-irs)i-qeq-j(-irs)l < kirq,

usJ -(14)

where ki is a constant because the function t -+ (it)-Pep(it) is obviously bounded on JR(p ~ 0). Since rp E S(Pn) it follows from (11)-(14) that each qth order derivative of F with respect to u1 , ... , Un is bounded in a punctured ball 0 < lui < t. Thus we have proved F E £(JRn). That F is rapidly decreasing is now clear from (7) and (11). Finally, if f is the function in S(JRn) whose Fourier transform is F then

J(sw) = F(sw) = /_: rp(w, r)e-irs dr;

hence by (4), f = rp and the theorem is proved.

To make further progress we introduce some useful notation. Let Sr ( x) denote the sphere {y : IY - x! = r} in JRn and A(r) its area. Let Br(x) denote the open ball {y : iY - xi < r }. For a continuous function f on Sr(x) let (Mr f)(x) denote the mean value

(Mr f)(x) = A(1 ) { f(w) dw, r Js~(x)

where dw is the Euclidean measure. Let K denote the orthogonal group O(n), dk its Haar measure, normalized by J dk = 1. If y E JRn, r = iYI then

(15) (Mr f)(x) = [ f(x + k · y) dk.

(Fig. 1.3) In fact, for x, y fixed both sides represent rotation-invariant functionals on C(Sr(x)), having the same value for the function f = 1. The rotations being transitive on Sr(x), (15) follows from the uniqueness of such invariant functionals. Formula (3) can similarly be written

(16) ~(x) = L rp(x + k. eo) dk

x+k·y

FIGURE !.3.


if {0 is some fixed hyperplane through the origin. We see then that if f ( x) = O{lxl-n), Ok the area of the unit sphere in JR.k, i.e., Ok = 2r(~J;),

(j)v (x) = [fcx + k · {0 ) dk = [(l/(x + k · y) dm(y)) dk

= { (MIYIJ)(x) dm(y) = On_1 1~n-z (~ { f(x + rw) dt) dr Jf.o 0 n Jsn-1 J

so

(17) ~v On-11 1 (f) (x) = ---n- lx- Yi- f(y) dy. Hn R"

We consider now the analog of Theorem 2.4 for the transform t.p -+ <{;. But t.p E SH(Pn) does not imply <{; E S(R.n). (If this were so and we by Theorem 2.4 write t.p = f, f E S(R.n) then the inversion formula in Theorem 3.1 for n = 3 would imply J f(x) dx = 0.) On a smaller space we shall obtain a more satisfactory result.

Let S* (R.n) denote the space of all functions f E S (R.n) which are orthogonal to all polynomials, i.e.,

{ f(x)P(x) dx = 0 for all polynomials P. }R,.

Similarly, let S*(Pn) C S(Pn) be the space of t.p satisfying

L t.p(w, r)p(r) dr = 0 for all polynomials p.

Note that under the Fourier transform the space S* (R.n) corresponds to the subspace S0 (R.n) c S(R.n) of functions all of whose derivatives vanish at 0.

Corollary 2.5. The transforms f -+ j, t.p -+ <{; are bijections of S* (R.n) onto S* (Pn) and of S* (Pn) onto S* (R.n), respectively.

The first statement is clear from (8) if we take into account the elementary fact that the polynomials x -+ (x, w)k span the space of homogeneous polynomials of degree k. To see that t.p -+ <{; is a bijection of S* (Pn) onto S* (R.n ) we use ( 17), knowing that t.p = j for some f E S* (lll" ) . The right hand side of (17) is the convolution of f with the tempered distribution lxl-1 whose Fourier transform is by Chapter V, §5 a constant multiple of lul1-n. (Here we leave out the trivial case n = 1.) By Chapter V, (12) this convolution is a tempered distribution whose Fourier transform is a constant multiple of !ul1-n Jcu). But, by Lemma 5.6, Chapter V this lies in the space So (R.n) since J does. Now (17) implies that <{; = (J) v E S* (R.n) and that <{; ~ 0 if t.p ~ 0. Finally we see that the mapping t.p -+ <{; is surjective because the function


FIGURE 1.4.

(where cis a constant} runs through S0 (JR.n) as f runs through S*(JR.n}. We now turn to the space V(JR.n) and its image under the Radon trans

form. We begin with a preliminary result. (See Fig. 1.4.)

Theorem 2.6. {The support theorem.) Let f E C(JR.n) satisfy the following conditions:

{i) For each integer k > 0, lxlk f(x) is bounded.

{ii) There exists a constant A > 0 such that

f(f.) = 0 for d(O, f.) > A,

d denoting distance.

Then

f(x) = 0 for !xl >A.

Proof. Replacing f by the convolution 'P* f where cp is a radial coo function with support in a small ball Be(O) we see that it suffices to prove the theorem for f E £(JR.n ). In fact, cp * f is smooth, it satisfies (i) and by (5) it satisfies (ii) with A replaced by A+€. Assuming the theorem for the smooth case we deduce that support (cp *f) C BA+e(O) so letting € -7 0 we obtain support (f) C Closure BA(O).

To begin with we assume f is a radial function. Then f(x) = F(lxl) where FE £(JR.) and even. Then fhas the form f(f.} = F(d(o,e}) where F is given by

F(p) = L .. -l F((p2 + IYI2)1/2} dm(y)' (p ~ 0)

because of the definition of the Radon transform. Using polar coordinates in JR.n-1 we obtain

(18) F(p) = On-1 1oo F((p2 + t2)1f2)tn-2 dt.

§2 The Radon Transform ... The Support Theorem 11

Here we substitute s = (p2 + t2)-112 and then put u = p-1 . Then (18) becomes

un-3 F(u-1) = !ln-1 lu. (F(s-1)s-n)(u2- s2)(n-3)/2 ds.

We write this equation for simplicity

(19) h(u) = lu. g(s)(u2 - s2)(n-S)/2 ds.

This integral equation is very close to Abel's integral equation (WhittakerWatson (1927], Ch. IX) and can be inverted as follows. Multiplying both sides by u(t2 - u2)(n-3)12 and integrating over 0::::; u::::; t we obtain

1t h(u)(t2- u2)(n-3)/2udu

= 1t [lou g(s)[(u2- s2)(t2- u2)J(n-3)/2 ds] u du

= 1t g(s) [i~s u[(t2- u2)(u2- s2)](n-3)/2 du] ds.

The substitution (t2 - s2)V = (t2 + s2) - 2u2 gives an explicit evaluation of the inner integral and we obtain

lot h(u)(t2 - u2)(n-3)12udu =Clot g(s)(t2 - s2)n-2 ds,

1 where C = 21-n?T-2T((n - 1)/2)/r(n/2). Here we apply the operator d(~2) = f£1£ (n- 1) times whereby the right hand side gives a constant multiple of r 1g(t). Hence we obtain

(20) F(C1 )en= ct -- (t2 - u2)(n-3)12un-2 F(u-1 ) du [ d ] n-11t d(t2) 0

where c-1 = (n- 2)!!ln/2n. By assumption (ii) we have F(u- 1) = 0 if u-1 2 A, that is if u::::; A-1. But then (20) implies F(t-1) = 0 if t::::; A-1, that is if r 1 2 A. This proves the theorem for the case when f is radial.

We consider next the case of a general f. Fix x E lRn and consider the function

9z (y) = L f( X + k · y) dk

as in (15). Then 9x satisfies (i) and

(21)

12 Chapter I. The Radon Transform on .R"

x + k · ~ denoting the translate of the hyperplane k · ~ by x. The triangle inequality shows that

d(O, x + k · ~) ~ d(O, ~) - jxj , x E R.n, k E K.

Hence we conclude from assumptions (i) and (21) that

(22) Uz(~) = 0 if d(O,~) >A+ jxj.

But 9z is a radial function so {22) implies by the first part of the proof that

(23) [ f(x + k · y) dk = 0 if IYI > A+ lxl.

Geometrically, this formula reads: The surface integral off over S1111(x) is 0 if the ball B1111(x) contains the ball BA(O). The theorem is therefore a consequence of the following lemma.

Lemma 2.7. Let f E C(Jlrl) be such that for each integer k > 0,

Suppose f has surface integral 0 over every sphere S which encloses the unit ball. Then f(x) = 0 for lxl > 1.

Proof. The idea is to perturb S in the relation

(24) Is f(s) dw(s) = 0

slightly, and differentiate with respect to the parameter of the perturbations, thereby obtaining additional relations. (See Fig. 1.5.) Replacing, as above, f with a suitable convolution cp * f we see that it suffices to prove the lemma for f in t"(R.n). Writing S = SR(x) and viewing the exterior of the ball BR(x) as a union of spheres with center x we have by the assumptions,

r J(y) dy = r J(y) dy, JBa(z) JR,.

FIGURE 1.5.


which is a constant. Differentiating with respect to Xi we obtain

(25) { (od)(x+y)dy=O. J BR(O)

We use now the divergence theorem

{26) { (divF)(y) dy = { (F, n)(s) dw(s) J BR(O) J SR(O)

for a vector field F on JR.n, n denoting the outgoing unit normal and dw the surface element on SR(O). For the vector field F(y) = f(x + y) 8~, we obtain from (25) and {26), since n = R-1(s1, ... , sn),

(27)

But by (24)

so by adding

This means that the hypotheses of the lemma hold for f(x) replaced by the function xif(x). By iteration

Is f(s)P(s) dw(s) = 0

for any polynomial P, so f = 0 on S. This proves the lemma as well as Theorem 2.6.

Corollary 2.8. Let f E C(JR.n) satisfy {i) in Theorem 2.6 and assume

for all hyperplanes e disjoint from a certain compact convex set C. Then

(28) f ( x) = 0 for x ~ C .

In fact, if B is a closed ball containing C we have by Theorem 2.6, f(x) = 0 for x ¢B. But Cis the intersection of such balls so (28) follows.


Remark 2.9. While condition (i) of rapid decrease entered in the proof of Lemma 2.7 (we used lxlk f(x) E L1 (1Rn) for each k > 0) one may wonder whether it could not be weakened in Theorem 2.6 and perhaps even dropped in Lemma 2.7.

As an example, showing that the condition of rapid decrease can not be dropped in either result consider for n = 2 the function

f(x,y) = (x +iy)-5

made smooth in JR2 by changing it in a small disk around 0. Using Cauchy's theorem for a large semicircle we have ft f(x) dm(x) = 0 for every line l outside the unit circle. Thus (ii) is satisfied in Theorem 2.6. Hence (i) cannot be dropped or weakened substantially.

This same example works for Lemma 2.7. In fact, let S be a circle lz - zol = r enclosing the unit disk. Then dw(s) = -ir z~~o so, by expanding the contour or by residue calculus,

(the residue at z = 0 and z = z0 cancel) so we have in fact

Is f(s) dw(s) = 0.

We recall now that DH(Pn) is the space of symmetric C00 functions cp(e) = cp(w,p) on pn oVcompact support such that for each k E z+, JR cp(w,p)pk dp is a homogeneous kth degree polynomial in w1 , ... ,wn. Combining Theorems 2.4, 2.6 we obtain the following characterization of the Radon transform of the space 'D(!Rn). This can be regarded as the analog for the Radon transform of the Paley-Wiener theorem for the Fourier transform (see Chapter V).

Theorem 2.10. {The Paley- Wiener theorem.) The Radon transform is a bijection of D(!Rn) onto DH(Pn).

We conclude this section with a variation and a consequence of Theorem 2.6.

Lemma 2.11. Let f E Cc(!Rn),A > 0, wo a fixed unit vector and N C S a neighborhood of wo in the unit sphere S C !Rn . Assume

i(w,p) = 0 for w E N, p > A.

Then

(29) f(x) = 0 in the half-space (x,wo) >A.

§3 The Inversion Formula 15

Proof. Let B be a closed ball around the origin containing the support of I. Let e > 0 and let He be the union of the half spaces (x,w) >A+ e as w runs through N. Then by our assumption

(30)

Now choose a ball Be with a center on the ray from 0 through -w0 , with the point (A+ 2e)w0 on the boundary, and with radius so large that any hyperplane~ intersecting B but not Be must be in He. Then by (30)

fW = 0 whenever ~ E pn, ~nEE= 0.

Hence by Theorem 2.6, l(x) = 0 for x ~ BE. In particular, l(x) = 0 for (x, w0 ) > A+ 2e; since e > 0 is arbitrary, the lemma follows.

Corollary 2.12. Let N be any open subset of the unit sphere sn-l. If f E Cc(lR.n) and

[(w,p) = 0 for p E JR.,w EN

then

Since [( -w, -p) = [(w,p) this is obvious from Lemma 2.11.

§ 3 The Inversion Formula

We shall now establish explicit inversion formulas for the Radon transform I -t J and its dual cp -t ij;.

Theorem 3.1. The function I can be recovered from the Radon transform by means of the following inversion formula

(31)

provided f(x) = O(lxi-N) for some N > n. Here c is the constant

c = (4n)<n-I)/2 r(n/2)jr(l/2).

Proof. We use the connection between the powers of L and the Riesz potentials in Chapter V, §5. Using (17) we in fact have

(32)


By Chapter V, Proposition 5.7, we thus obtain the desired formula (31). For n odd one can give a more geometric proof of (31). We start with

some general useful facts about the mean value operator Mr. It is a familiar fact that iff E C2 (JR.n) is a radial function, i.e., f(x) = F(r),r = lxl, then

(33) (Lf)(x) = ~d ~ + n- 1 ddF. r r r

This is immediate from the relations

EPf EJ2f (or ) 2 of o2r oxr = or2 OXi + or oxr .

Lemma 3.2. (i) LMr = Mr L for each r > 0.

{ii) For f E C2 (JR.n) the mean value (Mr f)(x) satisfies the "Darboux equation"

Lx((Mr f)(x)) = (!: + n ~ 1!) (Mr f(x)),

that is, the function F(x,y) = (MiYi!)(x) satisfies

Lx(F(x,y)) = Ly(F(x,y)).

Proof. We prove this group theoretically, using expression (15) for the mean value. For z E JR.n, k E K let Tz denote the translation x -7 x + z and Rk the rotation x -7 k · x. Since L is invariant under these transformations, we have if r = IYI,

(LMr f)(x) = L Lx(f(x + k · y) dk) = L (Lf)(x + k · y) dk = (Mr Lf)(x)

= [[(Lf) o Tx o Rk](y) dk = [[L(f o Tx o Rk)](y) dk

= Ly ([f(x+k·y)dk)

which proves the lemma. Now suppose f E S(JR.n ). Fix a hyperplane ~0 through 0, and an isom

etry g E M(n). As k runs through O(n), gk · ~0 runs through the set of hyperplanes through g · 0, and we have

<P(g · 0) = [ cp(gk · ~o) dk

and therefore

(j)v(g·O) = L(fe0

f(gk·y)dm(y))dk

= { dm(y) { f(gk·y)dk= { (MiYi!)(g·O)dm(y). leo lK leo


Hence

(34)

where On_1 is the area of the unit sphere in JRn-1 . Applying L to (34), using (33) and Lemma 3.2, we obtain

(35) L((J)v) = On-1 100 ( ~r~ + n ~ 1 ~~) rn-2 dr

where F(r) = (Mr f)(x). Integrating by parts and using

F(O) = f(x), lim rk F(r) = 0, r-too

we get

(( ~)v) { -On-d(x) if n = 3, L f = -On-1(n- 3) J0

00 F(r)rn-4 dr (n > 3).

More generally,

( {

00 r k ) {-(n-2)f(x) if k = 1' Lx ]0 (M J)(x)r dr = -(n-1- k)(k-1) Jt F(r)rk-2 dr, (k > 1).

If n is odd the formula in Theorem 3.1 follows by iteration. Although we assumed f E S(JRn) the proof is valid under much weaker assumptions.

Remark 3.3. The condition f(x) = O(lxi-N) for some N > n cannot in general be replaced by f(x) = O(lxl-n). In [1982) Zalcman has given an example of a smooth function f on JR2 satisfying f(x) = O(lxl-2 ) with

f(e) = 0 for all lines e and yet f =I 0. The function is even f(x) = O(lxj-3)

on each line which is not the x-axis. See also Armitage and Goldstein [1993).

Remark 3.4. It is interesting to observe that while the inversion formula requires f(x) = O(lxi-N) for one N > n the support theorem requires f(x) = O(lxi-N) for all N as mentioned in Remark 2.9.

We shall now prove a similar inversion formula for the dual transform <p -+ ip on the subspace S* (Pn).

Theorem 3.5. We have

c<p = (-o)<n-1)/2(ip), <p E S*(Pn)'

where c is the constant (4rr)Cn-l)/2r(n/2)jr(1/2).

Here 0 denotes as before the operator ~ and its fractional powers are again defined in terms of the Riesz' potentials on the !-dimensional p-space.

18 Chapter I. The Radon Transform on R.n

H n is odd our inversion formula follows from the odd-dimensional case in Theorem 3.1 if we put f = cp and take Lemma 2.1 and Corollary 2.5 into account. Suppose now n is even. We claim that

(36) n.-1 - n.-1 .......

((-L)-r f)= (-O)-r f f E S*(JiP).

By Lemma 5.6 in Chapter V, ( -L)(n-1)/2 f belongs to S*(R.n ). Taking the !-dimensional Fourier transform of (( -L)<n-1)/2 f) we obtain

(( -L)<n-1)/2 ff(sw) = isln-1J(sw).

On the other hand, for a fixed w, p --+ f( w, p) is in S* (R.). By the lemma quoted, the function p--+ (( -O)<n-1)12J)(w,p) also belongs to S*(R.) and

its Fourier transform equals isln-1J(sw). This proves (36). Now Theorem 3.5 follows from (36) if we put in (36)

<p = g' f = (g) v ' g E S* (R.n) '

because, by Corollary 2.5, g belongs to S*(Pn) . Because of its theoretical importance we now prove the inversion theo

rem (3.1) in a different form. The proof is less geometric and involves just the one variable Fourier transform.

Let 1l denote the Hilbert transform

(1-lF)(t) = !:_ roo F(p) dp 1r J_oo t- P

FE S(R.)

the integral being considered as the Cauchy principal value (see Lemma 3.7 below). For <p E S(Pn) let A<p be defined by

(37) (A )( ) _ dp"-r<p(w,p) n odd, { d"-1

<p w,p - d"-1 1-lp dp"- 1 <p(w,p) n even.

Note that in both cases (A<p)( -w, -p) = (A<p)(w,p) so A<p is a function on pn.

Theorem 3.6. Let A be as defined by (97). Then

cf = (Af)v, f E S(R.n),

where as before

c = ( -41r)(n-l)/2r(n/2)/f(l/2).

Proof. By the inversion formula for the Fourier transform and by (4)

f(x) = (21r)-n r dJ.,; roo ( roo e-isp f(w,p) dp) eis(z,w) sn-1 ds }sn-1 Jo }_oo


which we write as

j(x) = (21r)-n f F(w,x) dw = (21r)-n f ~(F(w, x)+F( -w,x)) dw. }sn-1 }sn-1

Using f( -w,p) = f(w, -p) this gives the formula

(38) j(x) = H27r)-n lsn-1 dw /_: lsln-leis(x,w) ds i: e-isp f(w,p) dp.

If n is odd the absolute value on s can be dropped. The factor sn-1 can

be removed by replacing f(w,p) by ( -i)n-1 da:;n-_\ f(w,p). The inversion formula for the Fourier transform on JR. then gives

as desired.

In order to deal with the case n even we recall some general facts.

Lemma 3.7. LetS denote the Cauchy principal value

s : 1/J -t lim r 1/J(x) dx. e-tO Jlxl"?.e X

Then S is a tempered distribution and S is the function

- . { -7ri s > 0 S(s) = -1rzsgn(s) = . - 0 .

7rZ S <

Proof. It is clear that S is tempered. Also xS = 1 so

21r6 =I= (xS) = i(S)'.

But sgn' = 26 so S = -1risgn+G. ButS and sgn are odd soC= 0. This implies

{39) (1-lF)(s) = sgn(s)F(s).

For n even we write in (38), lsln-1 = sgn(s)sn-1 and then (38} implies

(40) f(x) =Co r dw r sgn(s)eis(x,w) ds r ann--11 f(w,p)e-isp dp' }sn-1 }lR }lR dp

where eo= ~(-i)n-1 (27r)-n. Now we have for each FE S(JR) the identity

l sgn(s)eist ( l F(p)e-ips dp) ds = 21r(1iF)(t).


In fact, if we apply both sides to ;j with 1/J E S(JR), the left hand side is by

(39)

L (L sgn(s)eistff(s)ds);f(t)dt

= L sgn(s)F(s)21r'lj;(s) ds = 21r(1£F)(1/;) = 27r(1£F)(;f).

Putting F(p) = d~",.-_\ f(w,p) in (40) Theorem 3.6 follows also for n even.

For later use we add here a few remarks concerning 1£. Let FE V have

support contained in ( -R, R). Then

-i1r(1£F)(t) = lim 1 F(p) dp = lim f F(p) dp e~O e<!t-p! t- p e~O }It- p

where I = {p : IPI < R, e < jt - pi}. We decompose this last integral

f F(p) dp = f F(p) - F(t) dp + F(t) f _!}£_. }It-p }I t-p }It-p

The last term vanishes for jtj >Rand all e > 0. The first term on the right

is majorized by

r IF(t)- F(p) I dp ~ 2Rsup IF'I· j!P!<R t-p

Thus by the dominated convergence theorem

lim (1£F)(t) = 0. !tl~oo

Also if J C (-R, R) is a compact subset the mapping F -+ 1£F is contin

uous from VJ into £(JR) (with the topologies in Chapter V, §1).

§4 The Plancherel Formula

We recall that the functions on pn have been identified with the functions

cp on sn-l X lR which are even: cp( -w, -p) = cp(w,p). The functional

(41) cp-+ r r cp(w,p) dw dp Jsn-1 JR.

is therefore a well defined measure on pn, denoted dw dp. The group M( n)

of rigid motions of Rn acts transitively on pn: it also leaves the measure

dw dp invariant. It suffices to verify this latter statement for the translations

§4 The Plancherel Formula 21

Tin M(n) because M(n) is generated by them together with the rotations

around 0, and these rotations clearly leave dJ..J dp invariant. But

(<p o T)(w,p) = <p(w,p + q(w, T))

where q(w, T) E JR. is independent of p so

J/(<poT)(w,p)ch..Jdp= /1 <p(w,p+q(w,T))ch..Jdp= // <p(w,p)dpch..J,

proving the invariance. In accordance with (49)-(50) inCh. V the fractional power Ok is defined

on S(Pn) by

(42) (-Ok)<p(w,p) = H1(~2k) l <p(w,q)IP- qj-2k-1 dq

and then the !-dimensional Fourier transform satisfies

(43)

Now, iff E S(JR.n) we have by (4)

[(w,p) = (27r)-1 / J(sw)eisp ds

and

(44) ( -0) "4 1 [(w,p) = (27r)-1 Lis!~ J(sw)eisp ds.

n-1-

Theorem 4.1. The mapping f -t o-4-f extends to an isometry of

L2 (JR.n) onto the space L~(sn-1 x JR.) of even functions in L 2 (sn-l x JR.), the measure on sn-1 X JR. being

~(27r)1-n dJ..; dp.

Proof. By (44) we have from the Plancherel formula on JR.

(27r) L I( -0) "41 f{w,p)l 2 dp = L lsln-1if(swW ds

so by integration over sn-l and using the Plancherel formula for f(x) -t

J(sw) we obtain

r lf(xW dx = H27r)l-n r 10"41 [(w,p)l 2 ch..Jdp. JR" Jsn-lxR

It remains to prove that the mapping is surjective. For this it would suffice

to prove that if <p E L2(sn-l x JR.) is even and satisfies

r r <p(w,p)(-o)"41 f(w,p)ch..Jdp=0 Jsn-1 JR


for all f E S(lRn) then <p = 0. Taking Fourier transforms we must prove that if 1/J E L2(sn-l x JR) is even and satisfies

(45) r r 1/J(w,s)lsl"21 J(sw)dsdw = 0 }sn-1 }R

for all f E S(JRn) then 1/J = 0. Using the condition 1/J(-w, -s) = 1/J(w,s) we see that

r ro 1 -Js .. -1 } -oo 1/J(w, s)ls!2(n-l) f(sw) ds dw

= r roo 1/J(w, t)itl~(n-l) f(tw) dtdw lsn-1 Jo

so ( 45) holds with lR replaced with the positive axis JR+ . But then the function

satisfies

so "\li = 0 almost everywhere, whence 1/J = 0. If we combine the inversion formula in Theorem 3.6 with ( 46) below we

obtain the following version of the Plancherel formula

§5 Radon Transform of Distributions

It will be proved in a general context in Chapter II (Proposition 2.2) that

(46)

for f E Cc(lRn), <p E C(Pn) if dt; is a suitable fixed M(n)-invariant measure on pn. Thus dt; = 'Y dw dp where 'Y is a constant, independent of f and <p. With applications to distributions in mind we shall prove ( 46) in a somewhat stronger form.

Lemma 5.1. Formula (46) holds (with [and ip existing almost anywhere) in the following two situations:

§5 Radon Transform of Distributions 23

(a) f E L1 (JRn) vanishing outside a compact set; cp E C(Pn)

(b) f E Cc(lRn), cp locally integrable.

Also de= 0~1 dwdp.

Proof. We shall use the Fubini theorem repeatedly both on the product ]Rn X sn-1 and on the product ]Rn = lR X JRn-1 • Since f E £ 1 (JRn) we have

for each wE sn-1 that f{w,p) exists for almost all p and

r f(x) dx = r f{w,p) dp. }Rn jR

We also conclude that f(w,p) exists for almost all (w,p) E sn-1 X lit Next

we consider the measurable function

We have

(x,w) -t j(x)cp(w, (w,x}) on JRn X sn-1 •

1sn-1 xRn lf(x)cp(w, (w, x} )I dw dx

= lsn-1 (Ln lf(x)cp(w,(w,x})idx) dw

= lsn-1 (L lff(w,p)lcp(w,p)j dp) dw,

which in both cases is finite. Thus f ( x) · cp( w, (w, x}) is integrable on JRn x sn-1 and its integral can be calculated by removing the absolute

values above. This gives the left hand side of ( 46). Reversing the integra

tions we conclude that <P(x) exists for almost all x and that the double

integral reduces to the right hand side of ( 46).

The formula ( 46) dictates how to define the Radon transform and its

dual for distributions (see Chapter V). In order to make the definitions

formally consistent with those for functions we would require 8(cp) = S(ip),

f. (f) = E(j) if S and E are distributions on JRn and pn, respectively. But

while f E V(JRn) implies j E V(Pn) a similar implication does not hold

for cp; we do not even have <P E S(JRn) for cp E V(Pn) so 8 cannot be

defined as above even if Sis assumed to be tempered. Using the notation

c (resp. V) for the space of C00 functions (resp. of compact support) and V' (resp. £') for the space of distributions (resp. of compact support) we

make the following definition.

Definition. For S E £' (JRn) we define the functional 8 by

8(cp) = S(<P) for cp E c(Pn);


for I: E V 1 (Pn) we define the functional :E by

:E(J) = I:(j) for f E V(lRn) .

Lemma 5.2. (i) For each I: E V'(Pn) we have :E E V'(llln).

{ii} For each S E [ 1 (JR.n) we have S E E' (Pn).

Proof. For A > 0 let VA (JR.n) denote the set of functions f E V(llln) with support in the closure of BA(O). Similarly let 'VA(Pn) denote the set of functions <p E V(Pn) with support in the closure of the "ball"

.8A(O) = {~ E pn: d(O,~) <A}.

The mapping off -r [from 'VA(llln) to VA(Pn) being continuous (with the topologies defined in Chapter V, § 1) the restriction of :E to each 1) A (JR.n ) is continuous so (i) follows. That S is a distribution is clear from (3). Concerning its support select R > 0 such that S has support inside BR(O). Then if cp(w,p) = 0 for !PI ~ R we have rp(x) = 0 for lxl ~ R whence S(cp) = S(rp) = 0.

Lemma 5.3. ForSE t 1 (JR1l ), I: E 'V1(Pn) we have

(LSf = OS, (DI:) v = L:E.

Proof. In fact by Lemma 2.1,

(LS)(cp) = (LS)(rp) = S(Lrp) = S((Dcp)v) = S(Dcp) = (DS)(cp).

The other relation is proved in the same manner.

We shall now prove an analog of the support theorem (Theorem 2.6) for distributions. For A> 0 let .BA(O) be defined as above and let supp denote support.

Theorem 5.4. LetT E t 1(llln) satisfy the condition

suppT C Cl(.BA(O)), (C£ = closure).

Then

supp(T) C Cl(BA(O)).

Proof. For f E V(llln), cp E V(Pn) we can consider the "convolution"

(J X cp)(e) = { f(y)cp(e- y) dy, }Rn


where for ~ E pn, ~ - y denotes the translate of the hyperplane e by -y. Then

In fact, if eo is any hyperplane through 0,

(fxcp)v(x) = r dk r f(y)cp(x+k·eo-y)dy JK JR.,. = r dk r f(x- y)cp(y + k. eo) dy = (! * ip)(x). JK JR.,.

By the definition of T, the support assumption on T is equivalent to

T(i{J)=O

for all cp E 'D(Pn) with support in pn_Cl(,BA(O)). Let € > 0, let f E V(JRn) be a symmetric function with support in Cl(B,(O)) and let cp = 'D(Pn) have support contained in pn- Cl(J3A+e(O)). Since d(O, e- y) ::; d(O, e)+ IYI it follows that f x cp has support in pn- Cl(J3A(0)); thus by the formulas above, and the symmetry of f,

But then

(! * Tf(cp) = (! * T)( ip) = 0,

which means that (f*Tfhas support in Cl(J3A+e(O)). But now Theorem 2.6 implies that f * T has support in Cl(BA+e)(O). Letting € --7 0 we obtain the desired conclusion, supp(T) C Cl(BA(O)).

We can now extend the inversion formulas for the Radon transform to distributions. First we observe that the Hilbert transform 1l can be extended to distributions T on lR of compact support. It suffices to put

1l(T)(F) = T(-1lF), FE 'D(JR).

In fact, as remarked at the end of §3, the mapping F ---+ 1lF is a continuous mapping of 'D(JR) into £(JR). In particular 1l(T) E 'D'(JR).

Theorem 5.5. The Radon transformS---+ S (S E £'(1Rn)) is inverted by the following formula

cS = (AS)v, S E £ 1(1Rn),

where the constant c = (-411")(n-l)/2r(n/2)jr(l/2). In the case when n is odd we have also

26 Chapter I. The Radon Transform on R"

Remark 5.6. Since S has compact support and since A is defined by means of the Hilbert transform the remarks above show that AS E 'D' (Pn) so the right hand side is well defined.

Proof. Using Theorem 3.6 we have

The other inversion formula then follows, using the lemma.

In analogy with f3 A we define the "sphere" a A in pn as

aA = {e E pn: d(O,e) =A}.

From Theorem 5.5 we can then deduce the following complement to Theorem 5.4.

Corollary 5.7. Suppose n is odd. Then if s E e'(Rn) I

supp(S) EaR => supp(S) E SR(O).

To see this let € > 0 and let f E 'D(Rn) have supp(f) C BR-e(O). Then supp f E f3R-e and since A is a differential operator, supp(Aj) C f3R-e· Hence

cS(f) = S((AJ)v) = S(Aj) = 0

so supp(S) n BR-e(O) "1- 0. Since e > 0 is arbitrary,

supp(S) n BR(O) = 0.

On the other hand by Theorem 5.4, supp(S) C BR(O). This proves the corollary.

Let M be a manifold and df..L a measure such that on each local coordinate patch with coordinates (t1, ... , tn) the Lebesque measure dt1, ... , dtn and df..L are absolutely continuous with respect to each other. If h is a function on M locally integrable with respect to df..L the distribution <p --+ J <ph df..L will be denoted Th.

Proposition 5.8. (a) Let f E L1 (1Rn) vanish outside a compact set. Then the distribution Tt has Radon transform given by

(47)

{b) Let <p be a locally integrable function on pn . Then

(48)


Proof. The existence and local integrability of 1 and if; was established during the proof of Lemma 5.1. The two formulas now follow directly from Lemma 5.1.

As a result of this proposition the smoothness assumption can be dropped in the inversion formula. In particular, we can state the following result.

Corollary 5.9. (n odd.) The inversion formula

cf = £(n-1)/2((j)v),

c = { -47r)(n-1)/2f(n/2)/f(l/2), holds for all f E £ 1 (Jlrl) vanishing outside a compact set, the derivative interpreted in the sense of distributions.

Examples. If J.L is a measure (or a distribution) on a closed submanifold S of a manifold M the distribution on M given by r.p -+ J.L( r.p!S) will also be denoted by J.L·

(a) Let 80 be the delta distribution f-+ f(O) on Rn. Then

so

(49)

the normalized measure On gn-1 considered as a distribution on gn-1 X Jll

(b) Let ~o denote the hyperplane Xn = 0 in Rn, and 8e0 the delta distribution r.p-+ r.p(~o) on pn, Then

so

(50)

(8eo)v (f)= r f(x) dm(x) leo

the Euclidean measure of ~o.

(c) Let XB be the characteristic function of the unit ball B C R.n . Then by (47),

28 Chapter I. The Radon Transform on lRn

(d) Let n be a bounded convex region in JR.n whose boundary is a smooth surface. We shall obtain a formula for the volume of n in terms of the areas of its hyperplane sections. For simplicity we assume n odd. The characteristic function xn is a distribution of compact support and (xnfis thus well defined. Approximating xn in the £ 2-norm by a sequence ('1/Jn) C 1J(!l) we see from Theorem 4.1 that 8~n-l)/2;j}.n(w,p) converges in the £ 2-norm on pn. Since

I ;J)(e)cp(e) de= I '1/J.n(x)i{;(x) dx

it follows from Schwarz' inequality that ;J;n ---t (xnf in the sense of distributions and accordingly a<n-l)/2;j;.n converges as a distribution to a<n-l)/2{(xnf). Since the £ 2 limit is also a limit in the sense of distributions this last function equals the £ 2 limit of the sequence a<n-l)/2;j;_n. From Theorem 4.1 we can thus conclude the following result:

Theorem 5.10. Let n c JR.n (n odd} be a convex region as above and V(!l) its volume. Let A(w,p) denote the (n - I)-dimensional area of the intersection of n with the hyperplane (x, w) = p. Then

(51)

§6

V(!l) = l(27r)l-n { { la<n-1)/2 A(w,p) 12 dpdw. 2 Jsn-1 JR. 8p(n-1)/2

Integration over d-planes. X-ray Transforms. The Range of the d-plane Transform

Let d be a fixed integer in the range 0 < d < n. We define the d-dimensional Radon transform f -t 1 by

(52) f<e) = h f(x) dm(x) ~ ad-plane.

Because of the applications to radiology indicated in § 7,b) the !-dimensional Radon transform is often called the X-ray transform. Since a hyperplane can be viewed as a disjoint union of parallel d-planes parameterized by JR.n-l-d it is obvious from (4) that the transform f -t 1 is injective. Similarly we deduce the following consequence of Theorem 2.6.

Corollary 6.1. Let f, g E C(JR.n) satisfy the rapid decrease condition: For each m > 0, lxlm f(x) and lxlmg(x) are bounded on JRn. Assume for the d-dimensional Radon transforms

whenever the d-plane e lies outside the unit ball. Then

f(x) = g(x) for lxl > 1.

§6 Integration over d-planes. X-ray Transforms. 29

We shall now generalize the inversion formula in Theorem 3.1. If cp is a continuous function on the space of d-planes in JRn we denote by ip the point function

ip(x) = r cp(~) dJ.t(~)' l~:e~

where J.t is the unique measure on the (compact) space of d-planes passing through x, invariant under all rotations around x and with total measure 1. If a is a fixed d-plane through the origin we have in analogy with (16),

(53) ip(x)= [cp(x+k·a)dk.

Theorem 6.2. The d-dimensional Radon transform in JRn is inverted by the formula

(54)

where c = (47r)d/2r(n/2)jr((n- d)/2). Here it is assumed that f(x) = O(jxj-N) for some N > n.

Proof. We have in analogy with (34)

(j)v(x) = [ (1 f(x+k·y)dm(y)) dk

= 1 dm(y) [f(x+k·y)dk= 1(MIYIJ)(x)dm(y).

Hence

so using polar coordinates around x,

(55)

The theorem now follows from Proposition 5.7 in Chapter V. As a consequence of Theorem 2.10 we now obtain a generalization, char

acterizing the image of the space V(JRn) under the d-dimensional Radon transform.

The set G(d, n) of d-planes in JRn is a manifold, in fact a homogeneous space of the group M(n) of all isometries of JRn. Let Gd,n denote the manifold of all d-dimensional subspaces (d-planes through 0) of JRn. The parallel translation of a d-plane to one through 0 gives a mapping 1r of


G( d, n) onto Gd,n· The inverse image 1r-1 (a) of a member a E Gd,n is naturally identified with the orthogonal complement a..L. Let us write

~ = (a, x") = x" +a if a= 1r(~) and x" = a..L n ~.

(See Fig. 1.6.) Then (52) can be written

(56)

J(x" +a) = if(x' + x") dx'.

For k E Z + we consider the polynomial

(57)

Pk(u) = f f(x)(x, u)k dx. jR,.

If u = u" E a..L this can be written

FIGURE 1.6.

r f(x)(x, u")k dx = 11 f(x' + x")(x"' u")k dx' dx" jR,. ,..L ,.

so the polynomial

P,.,,.(u") = f f(x" +a)(x" ,u")kdx" },..L

is the restriction to a..L of the polynomial P,.. In analogy with the space 'DH(Pn) in No. 2 we define the space

'D H ( G ( d, n)) as the set of coo functions

<p(~) = <p,.(x") = <p(x" +a) (if~= (a,x"))

on G(d, n) of compact support satisfying the following condition. (H) : For each k E z+ there exists a homogeneous kth degree polynomial

Pk on IRn such that for each a E Gd,n the polynomial

P,.,,.(u") = { <p(x" +a)(x",u")kdx", u" E a..L, },..L

coincides with the restriction Pk la..L.

Theorem 6.3. The d-dimensional Radon transform is a bijection of 'D(.!Rn) onto 'DH(G(d,n)) .


Proof. For d = n - 1 this is Theorem 2.10. We shall now reduce the case of general d :::; n - 2 to the case d = n - 1. It remains just to prove the surjectivity in Theorem 6.3.

We shall actually prove a stronger statement.

Theorem 6.4. Let cp E V(G(d, n)) have the property: For each pair u, r E Gd,n and each k E z+ the polynomials

P.,.,k(u) = f cp(x" +u)(x",u}kdx" }.,..l.

Pr,k(u) = { cp(y" + r)(y", u)k dy" lr.l.

agree for u E u.L n r.L. Then cp = f for some f E V(JRn).

Proof. Let cp = V( G( d, n)) have the property above. Let w E JRn be a unit vector. Let u, r E Gd,n be perpendicular tow. Consider the (n- d- I)dimensional integral

(58) W'.,.(w,p) = { cp(x11 + u)dn-d-l(x"), p E JR. J(w,x"}=p,x"Eo-.l.

We claim that

w.,.(w,p) = W'r(w,p).

To see this consider the moment

L w.,.(w,p)pk dp

= Lpk (/ cp(x" +u)dn-d-l(x")) dp= l.l. cp(x" +u)(x",w)kdx"

= f cp(y" + r)(y",w}k dy" = f W'r(w,p)pk dp. lr.l. JR

Thus w.,.(w,p)- W'r(w,p) is perpendicular to all polynomials in p; having compact support it would be identically 0. We therefore put W'(w,p) = w.,.(w,p). Observe that w is smooth; in fact for win a neighborhood of a fixed w0 we can let u depend smoothly on w so by (58), w.,.(w,p) is smooth.

Writing

we have

(x",w)k = 2::: Pa(x")w"', lal=k


where

Here Aa is independent of a if wEal..; in other words, viewed as a function of w, Aa has for each a a constant value as w varies in al.. n 81 (0). To see that this value is the san1e as the value on rl.. n 81 (0) we observe that there exists apE Gd,n such that pl.. n al.. f. 0 and pl.. n rl.. f. 0. (Extend the 2-plane spanned by a vector in al.. and a vector in rl.. to an ( n - d)plane.) This shows that Aa is constant on 81 (0) soW E VH(Pn). Thus by Theorem 2.10,

(59) w(w,p) = f f(x) dm(x) J(x,w)=p

for some f E V(JRn). It remains to prove that

(60) c.p(x" +a) = L f(x' + x") dx'.

But as x 11 runs through an arbitrary hyperplane in al.. it follows from (58) and (59) that both sides of (60) have the same integral. By the injectivity of the (n-d-1)-dimensional Radon transform on al.. equation (60) follows. This proves Theorem 6.4.

Theorem 6.4 raises the following elementary question: If a function f on JRn is a polynomial on each k-dimensional subspace, is f itself a polynomial? The answer is no fork = 1 but yes if k > 1. See Proposition 6.13 below, kindly communicated by Schlichtkrull.

We shall now prove another characterization of the range of V(JRn) under the d-plane transform (for d ~ n- 2). The proof will be based on Theorem 6.4:

Given any d + 1 points (xo, ... , xd) in general position let ~(x0 , ••• , xd) denote the d-plane passing through them. If c.p E t'(G(d, n)) we shall write c.p(xo, ... , xd) for the value c.p(~(xo, ... , xd)). We also write V( {xi -xoh=l,d) for the volume of the parallelepiped spanned by vectors (xi- x0 ), (1 ~ i ~ d). The mapping

(Al, · · · , Ad) -t Xo + :Ef=lAi(Xi - Xo)

is a bijection of JRd onto ~(x0 , ••• ,xd) and

The range V(JRn) can now be described by the following alternative to Theorem 6.4. Let xf denote the kth coordinate of Xi·


Theorem 6.5. Iff E V(!Rn) then cp = f satisfies the system

(62) (ai,kai,l- ai,kai,t)(cp(xo, ... ,xd)/V({xi- xo}i=l,d)) = 0,

where

o ~ i, i ~ d, 1 ~ k, e ~ n , ai,k = a 1 ax~ . Conversely, if cp E V(G(d, n)) satisfies {62} then cp = f for some f E V(!Rn).

The validity of (62) for cp = fis obvious from (61) just by differentiation under the integral sign. For the converse we first prove a simple lemma.

Lemma 6.6. Let cp E E(G(d,n)) and A E O(n). Let 1/J = cp o A. Then if cp(xo, ... , xd) satisfies {62} so does the function

1/J(xo, ... , xd) = cp(Axo, ... , Axd).

Proof. Let Yi = Axi so yf = :Epatpxf· Then, if Di,k = ajayf,

(63) (ai,kai,l- ai,kai,L) = :E;,q=l apkaqt(Di,pDj,q - Di,qDj,p).

Since A preserves volumes, the lemma follows.

Suppose now cp satisfies (62). We write (J' = ((}'1 , ... ,(J'd) if ((J'j) is an orthonormal basis of (J'. If x11 E (J'.l, the (d +I)-tuple

( Ill/ II) X , X + (J'l, •.. , X + (J' d

represents the d-plane x" + (J' and the polynomial

(64)Po-,k(u") = f cp(x"+(J'){x",u"}kdx" lo-.l.

= { cp(x", x" +(}'1 , ..• , x" +(J' d) (x", u")k dx" , u" E (J'.l , lo-.l.

depends only on (J'. In particular, it is invariant under orthogonal transformations of ((}'1 , ••. , (J'd)· In order to use Theorem 6.4 we must show that for any (J', T E Gd,n and any k E z+,

(65) Po-,k(u) = Pr,k(u) for u E (J'.l n T.l, lui= 1.

The following lemma is a basic step towards (65).

Lemma 6.7. Assume cp E G(d,n) satisfies {62). Let

(J' = ((J'l,··· ,(J'd),r = (rl,··· ,rd)

be two members of Gd,n· Assume

(J'i = Tj for 2 ~ j ~ d.

Then


Proof. Let ei(l ~ i ~ n) be the natural basis of Rn and e = (e1 , ••. ,ed). Select A E 0 ( n) such that

o- = Ae , u = Aen .

Let

1J = A-1r = (A-1r1, ... ,A-1rd) = (A-1r1,e2, ... ,ed).

The vector E = A-1 r1 is perpendicular to ei (2 ~ j ~d) and to en (since u E rL). Thus

n-1

E = a1e1 + L aiei d+l

( ai + La: = 1) . i

In (63) we write P:,k for Pu,k· Putting x" = Ay and 'ljJ = cp o A we have

and similarly

Thus, taking Lemma 6.6 into account, we have to prove the statement:

(66)

where e = (el,··· ,ed),1J = (E,e2 , ••• ,ed), E being any unit vector perpendicular to ei (2 ~ j ~d) and to en. First we take

E = Et = sinte1 + costei (d < i < n)

and put €t = (Et, e2, ... , ed). We shall prove

(67)

Without restriction of generality we can take i = d + 1. The space ef consists of the vectors

n

(68) Xt = (-coste1 +sinted+l)..\d+l + L Aiei, Aj E JR. i=d+2

Putting P(t) = Pf,,k(en) we have

(69) P(t) = r cp(xt, Xt + Et, Xt + €2, ... 'Xt + ed)A~ dAn ... d..\d+l. }JRn-d


In order to use (62) we replace t.p by the function

'lj;(xo, ... ,xd)=t.p(xo, ... ,xd)/V({xi-xo}i=l,d)·

Since the vectors in (68) span volume 1 replacing t.p by 'lj; in (68) does not change P(t). Applying 8j8t we get (with d)..= dA.n ... dA.d+l),

d

(70) P' (t) = l .. -d [ {; Ad+l (sin t 8j,l 'lj; +cost 8i,d+l 'lj;)

+ cost 82,1 'lj; - sin t 82,d+1 'lj;] A.! d).. .

Now t.p is a function on G(d,n). Thus for each i,j it is invariant under the substitution

Yk = Xk (k #- i), Yi = SXi + (1- s)Xj = Xj + s(Xi- Xj), S > 0

whereas the volume changes by the factors. Thus

Taking 8/ 8s at s = 1 we obtain

n

(71) 'lj;(xo, ... , xd) + L(xf- x1)(8i,k¢)(xo, ... , xd) = 0. k=l

Note that in (70) the derivatives are evaluated at

(72) (xo, ... , xd) = (xt, Xt + Et, Xt + e2, ... , Xt + ed).

Using (71) for (i,j) = (1,0) and (i,j) = (0,1) and adding we obtain

(73) sint (8o,l¢ + 81,1¢) +cost (8o,d+l¢ + 81,d+1'1j;) = 0.

For i ~ 2 we have

and this gives the relations

(74) 'lj;(xo, ... , xd) + (8i,i¢)(xo, ... , Xd) = 0, (75) 'lj;- sin t (8i,I¢) -cost (8i,d+l 'lj;) + 8i,i¢ = 0.

Thus by (73)-(75) formula (70) simplifies to

P'(t) = l,._)cost (~,I¢)- sint (82,dH¢)JA! d)...


In order to bring in 2nd derivatives of tf; we integrate by parts in An,

Since the derivatives Oj,ktP are evaluated at the point (72) we have in (76)

(77)

and also, by (68) and (72),

(78)

We now plug (77) into (76) and then invoke equations (62) for tf; which give

d d d d

(79) L Oi,n02,1 t/J = 02,n L Oi,l t/J, L Oi,n02,d+l t/J = 02,n L Oi,d+l t/J: 0 0 0 0

Using (77) and (79) we see that (76) becomes

-(k + l)P'(t) =

{ [~,n(cost:Ei8i,ltP-sintEi8i,d+ltP)] (xt,Xt+Et, ... ,xt+ed)..\~+1 d..\ }R.n-d

so by (78)

Since d+ 1 < n, the integration in ..\d+l shows that P'(t) = 0, proving (67). This shows that without changing P,,k(en) we can pass from e =

( e1, ... , ed) to

By iteration we can replace e1 by

but keeping e2, ... , ed unchanged. This will reach an arbitrary E so (66) is proved.


We shall now prove (65) in general. We write a and r in orthonormal bases, a = (a1, ... , ad), r = (r1, ... , rd)· Using Lemma 6.7 we shall pass from a tor without changing Pa-,k(u), u being fixed.

Consider r1. If two members of a, say aj and ak, are both not orthogonal to r1 that is ((ail rt) "# 0, (ak, r1) "# 0) we rotate them in the (aj, ak)-plane so that one of them becomes orthogonal to r1. As remarked after (63) this has no effect on Pa-,k(u). We iterate this process (with the same r1) and end up with an orthogonal frame (ai, ... , a;t) of a in which at most one entry a; is not orthogonal to r1. In this frame we replace this a; by r1. By Lemma 6.7 this change of a does not alter Pa-,k(u).

We now repeat this process with r2 ,r3 .•• , etc. Each step leaves Pa-,k(u) unchanged {and u remains fixed) so this proves (65) and the theor':.._m.

We consider now the cased= 1, n = 3 in more detail. Here f -t f is the X-ray transform in JR3 . We also change the notation and write~ for x0, 1J for x1 so V({x1 - xo}) equals I~ -7JI· Then Theorem 6.5 reads as follows.

Theorem 6.8. The X-ray transform f -t fin IR3 is a bijection of 'D(IR3 )

onto the space of <p E 'D( G(1, 3)) satisfying

(80) (~~- ~~) (<p(~,7])) = 0, 1 ~ k,£ ~ 3. a~k a1Jt a~~. a1Jk 1~ - 111

Now let G' {1, 3) C G(1, 3) denote the open subset consisting of the nonhorizontal lines. We shall now show that for <p E 'D(G(1, n)) (and even for <p E £(G'(1,n))) the validity of (80) for (k,£) = (1,2) implies (80) for general (k,£). Note that (71) (which is also valid for <p E £(G'(1,n))) implies

cp(~,1J) + ~(~i-1Ji)_?_ (<p(~,7J)) =0. I~ - 111 "Y a~i I~ - 11!

Here we apply a I a1Jk and obtain

(~ a2 a a ) (cp(~,7J)) f;;t(~i -1Ji) a~ia1Jk - a~k + a1Jk 1~ -171 = 0 ·

Exchanging ~ and 1J and adding we derive

(81) tee- -1]-> (~-~) (~c~,1J)) -o i=l • • at;.ia1Jk at;.ka1Ji It; - 11! -

fork= 1, 2, 3. Now assume (80) for (k, £) = (1, 2). Taking k = 1 in (81) we derive (80) for (k,£) = (1,3). Then taking k = 3 in (81) we deduce (80) for ( k, £) = ( 3, 2). This verifies the claim above.

We can now put this in a simpler form. Let £(1;., 17) denote the line through the points t;, "# 1J· Then the mapping


is a bijection of JR4 onto G'(1,3). The operator

{)2 02 A=-----

o6o1J2 o6o"11 (82)

is a well defined differential operator on the dense open set G' (1, 3). If <p E E ( G ( 1, 3)) we denote by '1/J the restriction of the function ( e, 1J) -t <p(e,'TJ)/le -TJI to G'(1,3). Then we have proved the following result.

Theorem 6.9. The X-ray transform f -t f is a bijection of D(JR3 ) onto the space

(83) {<p E D(G(1,3)): Atf; = 0}.

We shall now rewrite the differential equation (83) in PlUcker coordinates. The line joining e and 1J has Plucker coordinates (p1, P2, Pa, q1, q2, q3) given by

i j e1 6 "11 "12

which satisfy

(84)

k 6 "13

= P1i + p~ + p3k, qi = I ei 11 T}i

Conversely, each ratio <PI : P2 : P3 : q1 : q2 : q3) determines uniquely a line provided (84) is satisfied. The set G'(1, 3) is determined by q3 =f. 0. Since the common factor can be chosen freely we fix q3 as 1. Then we have a bijection T : G' (1, 3) -t JR4 given by

(85) x1 = P2 + q2, x2 = -pl- q1, xa = P2- q2, X4 = -p1 + q1

with inverse

(pl,P2,P3, q1, q2) = GC-xz-x4), tCx1 +x3), :t(-xi-x~+x~+x~), t<-xz+x4), ~(x1-x3)).

Theorem 6.10. If <p E D(G(1, 3)) satisfies {83) then the restriction <pjG'(1, 3) (with q3 = 1) has the form

(86) <p(e,'T}) = le- "11 u(P2 + qz, -pl- ql,P2- q2, -pl + ql)

where u satisfies

o2u o2u o2u o2u -+------0 oxi ox~ ox~ ox~ - .

(87)

On the other hand, if u satisfies {87) then {86} defines a function <p on G'(1,3) which satisfies {80}.


Proof. First assume t.p E V(G(1, 3)) satisfies (83) and define u E £(JR4 ) by

(88)

where l E G'(1, 3) has Pliicker coordinates (pi.p2,p3, q1, q2, 1). On the line l consider the points e,'f/ for which 6 = O,'f/a = -1 (so qa = 1). Then since

Pt = -6, P2 = 6, q1 = 6- 'f/1. q2 = 6- 'f/2

we have

t.p(e,'f/) (89) le-'fll =u(el +6 -TJ2, -e1 +6 -T]l, e1-6 +TJ2, e1 +6 -TJl).

Now (83) implies (87) by use of the chain rule. On the other hand, suppose u E £(JR4 ) satisfies (87). Define t.p by (88).

Then t.p E £(G'(1, 3)) and by (89),

A (t.p(e, TJ)) = o. 1e -111

As shown before the proof of Theorem 6.9 this implies that the whole system (80) is verified.

We shall now see what implications Asgeirsson's mean-value theorem (Theorem 4.5, Chapter V) has for the range of the X-ray transform. We have from (85),

(90) 121< u(r cos t.p, r sin t.p, 0, 0) dt.p = 12

1< u(O, 0, r cos t.p, r sin t.p) dt.p.

The first points (r cos t.p, r sin t.p, 0, 0) correspond via (85) to the lines with

(p ) ( r · r r 2 r · r 1) I.P2,Pa, q1, q2, qa = -2 sm t.p, 2 cost.p, - 4 , - 2 smt.p, 2 cos t.p,

containing the points Cet.6,6) = (¥cost.p,¥sint.p,O)

(TJt. T]2, TJa) = (¥(sin t.p +cos t.p), +¥(sin t.p - cos t.p ), -1)

with 1e -1712 = 1 + r42

• The points (0, 0, r cos t.p, r sin t.p) correspond via (85) to the lines with

containing the points

Cet,6,ea) = (~cost.p,¥sint.p,O)

(Tit, T]2, TJa) = (~(cos t.p- sin t.p), ¥(cos t.p +sin t.p), -1)

40 Chapter I. The Radon Transform on JRn

with I~ - 11! 2 = 1 + r42

• Thus (90) takes the form

(91) 127r cp{~ cosO,~ sin B, 0, HsinO +cos B), HsinO- cos 0), -1) dO

{21r = lo cp(~ cos 0, ¥sinO, 0, Hcos 0- sinO), ¥(cosO+ sin B), -1) dO.

The lines forming the arguments of cp in these integrals are the two families of generating lines for the hyperboloid (see Fig. 1.7)

Definition. A function cp E £(G'(1, 3)) is said to be a harmonic line function if

A(cp(~, 77)) = 0. I~ -111

Theorem 6.11. A function cp E £(G'(I, 3)) is a harmonic line function if and only if for each hyperboloid of revolution H of one sheet and vertical axis the mean values of cp over the two families of generating lines of H are equal. (The variable of integration is the polar angle in the equatorial plane of H.) FIGURE 1.7.

The proof of {91) shows that cp harmonic implies the mean value property for cp. The converse follows since (90) (with (0, 0) replaced by, an arbitrary point in JR2 ) is equivalent to (87) (Chapter V, Theorem 4.5).

Corollary 6.12. Let cp E V(G(l, 3)). Then cp is in the range of the Xray transform if and only if cp has the mean value property for arbitrary

hyperboloid of revolution of one sheet (and arbitrary axis}.

We conclude this section with the following result due to Schlichtkrull mentioned in connection with Theorem 6.4.

Proposition 6.13. Let f be a function on !Rn and k E z+, 1 < k < n.

Assume that for each k-dimensional subspace Ek C !Rn the restriction f!Ek

is a polynomial on Ek. Then f is a polynomial on !Rn.

Fork= 1 the result is false as the example f(x, y) = xyfx2 +y2 , f(O, 0) = 0 shows. We recall now the Lagrange interpolation formula. Let ao, ... , am

be distinct numbers in C. Then each polynomial P(x) (x E JR) of degree

§7 Applications 41

~ m can be written

P(x) = P(ao)Qo(x) + · · · + P(am)Qm(x),

where

m

Qi(x) =II (x- aJ)f(x- ai) II (ai- aj). j=O

In fact, the two sides agree at m + 1 distinct points. This implies the following result.

Lemma 6.14. Let f(xi, ... ,xn) be a function on JR.n such that for each i with Xj(j :f. i) fixed the function Xi -+ f(xb ... , Xn) is a polynomial. Then f is a polynomial.

For this we use Lagrange's formula on the polynomial XI --7

f(xi, xz, ... , Xn) and get

m

f(xi,··· ,xn) = 'L,f(aj,Xz, ... ,xm)Qj(XI). j=O

The lemma follows by iteration. For the proposition we observe that the assumption implies that f re

stricted to each 2-plane E2 is a polynomial on E2 . For a fixed (x2 , ••• ,xn) the point (xi,··· ,xn) is in the span of (1,0, ... ,0) and (O,xz, ... ,xn) so f(xb ... , Xn) is a polynomial in XI. Now the lemma implies the result.

§7 Applications

a) Partial differential equations.

The inversion formula in Theorem 3.1 is very well suited for applications to partial differential equations. To explain the underlying principle we write the inversion formula in the form

(92) f(x)=-yLT( f l<w,(x,w))dw). }g .. -1

where the constant 'Y equals !{21ri)I-n. Note that the function fw(x) = J(w, (x,w)) is a plane wave with normal w, that is, it is constant on each hyperplane perpendicular tow.

Consider now a differential operator

D = 'L,ak1 .•. k,.8f1 ••• 8~" (k)


with constant coefficients ak1> ... ,k,., and suppose we want to solve the differential equation

(93) Du=f

where f is a given function in S(JRn). To simplify the use of (92) we assume n to be odd. We begin by considering the differential equation

(94) Dv = fw,

where f w is the plane wave defined above and we look for a solution v which is also a plane wave with normal w. But a plane wave with normal w is just a function of one variable; also if v is a plane wave with normal w so is the function Dv. The differential equation (94) (with v a plane wave) is therefore an ordinary differential equation with constant coefficients. Suppose v = Uw is a solution and assume that this choice can be made smoothly in w. Then the function

(95) n-1 ls u = 'Y L -2- Uw dw sn-1

is a solution to the differential equation (93). In fact, since D and L "2 1

commute we have

n-1 ls n-1 ls Du = "(L_2_ Duw dw = "(L :r fw dw =f. sn-1 sn-1

This method only assumes that the plane wave solution Uw to the ordinary differential equation Dv = f w exists and can be chosen so as to depend smoothly on w. This cannot always be done because D might annihilate all plane waves with normal w. (For example, take D = {)2 fax 18x2

and w = (1, 0).) However, if this restriction to plane waves is never 0 it follows from a theorem of Treves [1963] that the solution uw can be chosen depending smoothly on w. Thus we can state

Theorem 7.1. Assuming the restriction Dw of D to the space of plane waves with normal w is =f. 0 for each w formula (95) gives a solution to the differential equation Du = f (f E S(Rn)).

The method of plane waves can also be used to solve the Cauchy problem for hyperbolic differential equations with constant coefficients. We illustrate this method by means of the wave equation !Rn,

(96) [)2u

Lu = at2 , u(x,O) = fo(x), Ut(x,O) = !1(0),

/o, h being given functions in V(Rn ).

Lemma 7.2. LethE C2 (JR) and wE sn-t. Then the function

v(x, t) = h( (x, w))

satisfies Lv = (82 fat2 )v.

§7 Applications 43

The proof is obvious. It is now easy, on the basis of Theorem 3.6, to write down the unique solution of the Cauchy problem (96).

Theorem 7.3. The solution to {96} is given by

(97) u(x, t) = fs.,._1 (Sf)(w, (x,w) + t) dw

where

Sf= { c(an-1Jo + an-2 h)'

c1i(an-1Jo + an-2 h)'

Here 8 = ofop and the constant c equals

c = ~(2rri) 1-n.

n odd

n even.

Lemma 7.2 shows that (97) is annihilated by the operator L- 82 j8t2 so we just have to check the initial conditions in (96).

(a) If n > 1 is odd then w --t (an-lfo)(w, (x,w}) is an even function on sn-1 but the other term in Sf, that is the function w--t (an-2h)(w, (x,w) ), is odd. Thus by Theorem 3.6, u(x,O) = fo(x). Applying ojot to (97) and putting t = 0 gives Ut(x,O) = fi(x), this time because the function w --t

(on Jo)(w, (x,w)) is odd and the function w --t (an-1 h)(w, (x,w)) is even.

{b) If n is even the same proof works if we take into account the fact that 1i interchanges odd and even functions on JR.

Definition. For the pair f = {fo,JI} we refer to the function Sf in (97) as the source.

In the terminology of Lax-Philips [1967] the wave u(x, t) is said to be

(a) outgoing if u(x, t) = 0 in the forward cone lxi < t;

(b) incoming if u(x, t) = 0 in the backward cone lxl < -t.

The notation is suggestive because "outgoing" means that the function x --t u(x, t) vanishes in larger balls around the origin as t increases.

Corollary 7.4. The solution u(x, t) to {96} is

{i) outgoing if and only if (S f)(w, 8) = 0 for 8 > 0, all w.

{ii) incoming if and only if (Sf)(w, 8) = 0 for 8 < 0, all w.

44 Chapter I. The Radon Transform on JRn

Proof. For (i) suppose (Sf)(w,s) = 0 for s > 0. For lxl < t we have (x, w} +t ~-!xi +t > 0 so by (97) u(x, t) = 0 sou is outgoing. Conversely, suppose u(x, t) = 0 for lxl < t. Let to > 0 be arbitrary and let cp(t) be a smooth function with compact support contained in (t0 , oo).

Then if lxl < to we have

0 = f u(x, t)cp(t) dt = { dw f (Sf)(w, (x, w} + t)cp(t) dt JfR Jsn-1 JfR

= fs,._1

dw l (Sf)(w,p)cp(p- (x,w)) dp.

Taking arbitrary derivative ak I axil 0 0 0 8xik at X = 0 we deduce

f ( f (Sf)(w,p)wil···wik)cakcp)(p)dp=O JfR Jsn-1 for each k and each cp E V( t0 , oo). Integrating by parts in the p variable we conclude that the function

(98) p-+ { (8/)(w,p)wit···Wikdw, pElR Jsn-1 has its kth derivative = 0 for p > t0 . Thus it equals a polynomial for p > to. However, if n is odd the function (98) has compact suppprt so it must vanish identically for p > to.

On the other hand, if n is even and F E V(JR) then as remarked at the end of §3, limltl-+oo(1iF)(t) = 0. Thus we conclude again that expression (98) vanishes identically for p > to.

Thus in both cases, if p > t0 , the function w -+ (Sf) ( w, p) is orthogonal to all polynomials on sn-l' hence must vanish identically.

One can also solve (96) by means of the Fourier transform

J(() = { f(x)e-i(x,t;) dx. JfR,.

Assuming the function x -+ u(x, t) in S(JRn) for a given t we obtain

iitt((,t) + ((,(}u((,t) = o.

Solving this ordinary differential equation with initial data given in (96) we get

(99) u((, t) = fo(() cos(j(jt) + f(() sinl~flt) .

The function (-+ sin(j(jt)/1(1 is entire of exponential type It! on en (of at most polynomial growth on JRn) so by Theorem 3.3, Chapter V there exists a Tt E £'(1Rn) with support in Blti(O) such that

sin(j(jt) = { -i(t;,x) d,., ( ) j(j JR,.. e .L t X .

§7 Applications 45

Theorem 7.5. Given Jo,h E £(Rn) the function

(100) u{x, t) = (fo * T;)(x) +(It* Tt)(x)

satisfies (96}. Here Tf stands for Ot(Tt)·

Note that (96) implies (100) if fo and It have compact support. The converse holds without this support condition.

Corollary 7.6. If fo and It have support in BR{O) then u has support in the region

lxl ~it! +R.

In fact, by (100) and support property of convolutions (Ch. V, §2), the function x -t u(x, t) has support in BR+Itl(o)-. While Corollary 7.6 implies that for fo, It E V(Rn) u has support in a suitable solid cone we shall now see that Theorem 7.3 implies that if n is odd u has support in a conical shell (see Fig. !.8).

/ FIGURE 1.8.

Corollary 7.7. Let n be odd. Assume fo and ft have support in the ball BR{O).

(i) Huygens' Principle. The solution u to (g6) has support in the conical shell

(101) jtj - R ~ lxl :5 jtj + R,

which is the union for IYI ~ R of the light cones,

Cy = {(x,t): jx-yj = jtj}.

(ii) The solution to {96) is outgoing if and only if

(102) fo(w,p)= i 00h(w,s)ds, p>O , allw


and incoming if and only if

~ [P ~ fo(w,p) =- 1-oo h(w,s)ds, p<O, allw.

To verify (i) note that since n is odd, Theorem 7.3 implies

(103) u(O,t) = 0 for jtj? R.

If z E !Rn, FE £(1Rn) we denote by pz the translated function y -t F(y+z). Then uz satisfies (96) with initial data !6, ft which have support contained in BR+Izi(O). Hence by {103)

(104) u(z,t) = 0 for jtj > R + lzl.

The other inequality in (101) follows from Corollary 7.6. For the final statement in (i) we note that if IYI :::::; R and (x , t) E Cy

then jx - yj = t so lxl :::=; lx - yj + IYI :::::; !tl + R and It! = jx - yj :::::; lxl + R proving (101). Conversely, if (x, t) satisfies (101) then (x, t) E Cy with y = x -ltl~ = ~(lxl- t) which has norm:::=; R.

For (ii) we just observe that since h(w,p) has compact support inp, (102) is equivalent to (i) in Corollary 7.4.

Thus (102) implies that fort> 0, u(x, t) has support in the thinner shell itl:::::; lxl:::::; !tl +R.

(b) X-ray Reconstruction.

The classical interpretation of an X-ray picture is an attempt at reconstructing properties of a 3-dimensional body by means of the X-ray projection on a plane.

In modern X-ray technology the picture is given a more refined mathematical interpretation. Let B c JR3 be a body (for example a part of a human body) and let f ( x) denote its density at a point x. Let ~ be a line in IR3 and suppose a thin beam of X-rays is directed at B along f Let Io and I respectively, denote the intensity of the beam before entering FIGURE !.9. B and after leaving B (see

Fig. 1.9). As the X-ray traverses the distance dx along~ it will undergo the relative intensity loss ill I I = f ( x) ilx. Thus dl I I = - f ( x) dx whence

§7 Applications 47

(105) log(l0 /I) = fet(x)dx,

the integral j( e) of f along e. Since the left hand side is determined by the X~ray picture, the X-ray reconstruction problem amounts to the determination of the function f by means of its line integrals [(e). The inversion formula in Theorem 3.1 gives an explicit solution of this problem.

If B0 C B is a convex subset (for example the heart) it may be of interest to determine the density off outside B0 using only X-rays which do not intersect Bo. The support theorem (Theorem 2.6, Cor. 2.8 and Cor. 6.1) implies that f is determined outside B0 on the basis of the integrals [(e) for which e does not intersect Bo. Thus the density outside the heart can be determined by means of X-rays which bypass the heart.

In practice one can of course only determine the integrals [(e) in (105} for finitely many directions. A compensation for this is the fact that only an approximation to the density f is required. One then encounters the mathematical problem of selecting the directions so as to optimize the approximation.

As before we represent the line e as the pair e = ( w, z) where w E R.n is a unit vector in the direction of e and z = e n w..L ( j_ denoting orthogonal complement). We then write

(106) f(e) = [(w, z) = (Pwf)(z).

The function Pwf is the X-ray picture or the radiograph in the direction w. Here f is a function on Rn vanishing outside a ball B around the origin and for the sake of Hilbert space methods to be used it is convenient to assume in addition that f E L 2 (B). Then f E L 1 (R.n) so by the Fubini theorem we have: for each w E sn-l, Pwf(z) is defined for almost all z E w..L. Moreover, we have in analogy with (4),

(107) i(() = { (Pwf)(z)e-i(z,,) dz (( E w..L). lw.l..

Proposition 7.8. An object is determined by any infinite set of radiographs.

In other words, a compactly supported function f is determined by the functions Pwf for any infinite set of w.

Proof. Since f has compact support f is an analytic function on Rn. But if!(() = 0 for ( E w..L we have J(TJ) = (w,TJ)9(TJ) (TJ E Rn) where g is also analytic. If Pw1 j, ... , Pwk f . .. all vanish identically for an infinite set w1 , ... ,w~c ... we see that for each k

k

J(TJ) = IT (wi, TJ)9k (TJ), i=l


where 9k is analytic. But this would contradict the power series expansion of J which shows that for a suitable w E sn-l and integer r ;::: 0, limHo f(tw)rr =I 0.

If only finitely many radiographs are used we get the opposite result.

Proposition 7.9. Let w1, ... ,wk E sn-l be an arbitrary finite set. Then there exists a function f E V(JRn ), f =¢ 0 such that

Pw; f = 0 for all 1 $ i $ k .

Proof. We have to find f E V(JRn ), f =¢ 0, such that J( () = 0 for ( E wf(l $ i $ k). For this let D be the constant coefficient differential operator such that

k

(Du)(17) =II (wi, 1J)U(1J) 1] E lin . 1

If u =¢ 0 is any function in V(JRn) then f = Du has the desired property.

We next consider the problem of approximate reconstruction of the function f from a finite set of radiographs Pw1 f, ... , Pw~of·

Let N; denote the null space of Pw; and let P; the orthogonal projection of L 2 (B) on the plane f + N;; in other words

(108)

where Q; is the (linear) projection onto the subspace Ni c L 2 (B). Put P = Pk ... P1. Let g E L 2 (B) be arbitrary (the initial guess for f) and form the sequence pmg, m = 1, 2, .... Let No= nfNi and let Po (resp. Qo) denote the orthogonal projection of L 2 (B) on the plane f + N0 (subspace N0). We shall prove that the sequence pmg converges to the projection Pog. This is natural since by Pog- f E No, Pog and f have the same radiographs in the directions W1J ••• , wk.

Theorem 7.10. With the notations above,

for each g E L 2 (B).

Proof. We have, by iteration of (108)

and, putting Q = Qk ... Q1 we obtain

§7 Applications 49

We shall now prove that Qm g --+ Qog for each g; since

Pog = Qo (g - f) + f

this would prove the result. But the statement about Qm comes from the following general result about abstract Hilbert space.

Theorem 7.11. Let 1l be a Hilbert space and Qi the projection of1l onto a subspace Ni C 1£(1 ~ i ~ k). Let No = n~ Ni and Qo : 1l --+ No the projection. Then if Q = Qk ... Ql

Qm g --+ Qog for each g E 1£, ·

Since Q is a contraction (IIQII ~ 1) we begin by proving a simple lemma about such operators.

Lemma 7.12. LetT: 1l--+ 1l be a linear operator of norm~ 1. Then

1l = Cf.((I- T)1l) EB Null space (I-T)

is an orthogonal decomposition, Cf. denoting closure, and I the identity.

Proof. If Tg = g then since liT* II= IITII ~ 1 we have

llull2 = (g,g) = (Tg,g) = (g,T*g) ~ IIYIIIIT*gll ~ llgll2

so all terms in the inequalities are equal. Hence

llu- T*gll 2 = llgll2 - (g,T*g)- (T*g,g) + IIT*gll2 = o

so T*g =g. Thus I-T and I-T* have the same null space. But (I -T*)g = 0 is equivalent to (g, (I- T)1l) = 0 so the lemma follows.

Definition. An operator T on a Hilbert space 1l is said to have property Sif

(109) llfnll ~ 1, IITfnll--+ 1 implies II(I- T)fnll--+ 0 ·

Lemma 7.13. A projection, and more generally a finite product of projections, has property (S).

Proof. If T is a projection then

whenever

llfnll ~ 1 and IITfnll--+ 1.


Let T2 be a projection and suppose T1 has property (S) and IITtll ~ 1. Suppose fn E 1i and llfnll ~ 1, IIT2Tdnll ---+ 1. The inequality implies IITdnll ~ 1 and since

11Tdnll2 = IIT2Tdnll 2 + II(I- T2)(Tdn)ll 2

we also deduce IITdnll ---+ 1. Writing

we conclude that T2T1 has property (S). The lemma now follows by induction.

Lemma 7.14. Suppose T has property (S) and IITII ~ 1. Then for each f E 1i

where 1r is the projection onto the fixed point space of T.

Proof. Let f E Ji. Since liT II ~ 1, IITn !II decreases monotonically to a limit a~ 0. If a= 0 we have Tnf---+ 0. By Lemma 7.12 1rT = T1r so 1rj = Tn1r f = 1rTn f so 1r f = 0 in this case. If a > 0 we put 9n = IITn fii- 1(Tn !). Then ll9nll = 1 and IITgnll-t 1. Since T has property (S) we deduce

Tn(I- T)f = IITnfii(I- T)gn---+ 0,.

Thus Tnh---+ 0 for all h in the range of I-T. If g is in the closure of this range then given t > 0 there exist h E (I - T)Ji such that 119 - hll < t. Then

whence Tng ---+ 0. On the other hand, if h is in the null space of I-T then Th = h so Tnh---+ h. Now the lemma follows from Lemma 7.12.

In order to deduce Theorem 7.11 from Lemmas 7.13 and 7.14 we just have to verify that No is the fixed point space of Q. But if Qg = g then

so equality signs hold everywhere. But the Qi are projections so the norm identities imply

which shows g E N0 • This proves Theorem 7.11.

Bibliographical Notes 51

BIBLIOGRAPHICAL NOTES

§§1-2. The inversion formulas

(i) f(x) = t{27ri)1-n L~n-l)/2 fsn-1 J(w, (w, x)), dw

(ii) f(x) = ~(21ri)-n L~n-2)/2 fsn-1 dw J::O :::~:;~) (n odd)

(n even)

for a function f E V(X) in terms of its plane integrals J(w,p) go back to Radon [1917] and John [1934], [1955]. According to Bockwinkel [1906] the case n = 3 had been proved before 1906 by H.A. Lorentz, but fortunately, both for Lorentz and Radon, the transformation f(x) --+ J(w,p) was not baptized "Lorentz transformation". In John [1955] the proofs are based on the Poisson equation Lu =f. Other proofs, using distributions, are given in Gelfand-Shilov [1960]. See also Nievergelt [1986]. The dual transforms, f --+ f, r.p --+ <p, the unified inversion formula and its dual,

were given by the author in [1964]. The second proof of Theorem 3.1 is from the author's paper [1959]. It is valid for constant curvature spaces as well. The version in Theorem 3.6 is also proved in Ludwig [1966].

The support theorem, the Paley-Wiener theorem and the Schwartz theorem (Theorems 2.4,2.6, 2.10) are from Helgason [1964], [1965a]. The example in Remark 2.9 was also found by D.J. Newman, cf. Weiss' paper [1967], which gives another proof of the support theorem. See also Droste [1983]. The local result in Corollary 2.12 goes back to John [1935); our derivation is suggested by the proof of a similar lemma in Flensted-Jensen [1977], p. 81. Another proof is in Ludwig [1966].See Palamodov and Denisjuk [1988] for a related inversion formula.

The simple geometric Lemma 2. 7 is from the authors paper [1965a] and is extended to hyperbolic spaces in [1980b). In the Proceedings containing [1966a] the author raised the problem (p. 174) to extend Lemma 2.7 to each complete simply connected Riemannian manifold M of negative curvature. If in addition M is analytic this was proved by Quinto [1993b) and Grinberg and Quinto [1998]. This is an example of injectivity and support results obtained by use of the techniques of microlocal analysis and wave front sets. As further samples involving very general Radon transforms we mention Boman [1990], [1992], [1993), Boman and Quinto [1987], [1993), Quinto [1983], [1992], [1993b], [1994a], [1994b), Agranovsky and Quinto [1996], Gelfand, Gindikin and Shapiro [1979).

Corollary 2.8 is derived by Ludwig [1966] in a different way: There he proposes ~ternative proofs of the Schwartz- and Paley-Wiener theorems by expanding f { w, p) in spherical harmonics in w. However, the principal point-the smoothness of the function Fin the proof of Theorem 2.4-is overlooked. Theorem 2.4 is from the author's papers [1964] [1965a].

Since the inversion formula is rather easy to prove for odd n it is natural to try to prove Theorem 2.4 for this case by showing directly that if r.p E SH(Pn) then the function f = cL(n-l)/2 ({p) for n odd belongs to S(lle) (in general <p ft S(JR")). This approach is taken in Gelfand-Graev-Vilenkin [1966], pp. 16-17. However, this method seems to offer some unresolved technical difficulties. For some generalizations see Kuchment and Lvin [1990), Aguilar, Ehrenpreis and Kuchment [1996] and Katsevich [1997]. Cor. 2.5 is stated in Semyanisty [1960].


§5. The approach to Radon transforms of distributions adopted in the text is from the author's paper [1966a]. Other methods are proposed in Gelfand-GraevVilenkin [1966] and in Ludwig [1966]. See also Ramm [1995].

§6. The d-plane transform and Theorem 6.2 are from Helgason [1959], p. 284. Formula (55) was already proved by Fuglede [1958]. The range characterization for the d-plane transform in Theorem 6.3 is from the 1980-edition of this book and was used by Kurusa [1991] to prove Theorem 6.5, which generalizes John's range theorem for the X-ray transform in R3 [1938]. The geometric range characterization (Corollary 6.12) is also due to John [1938]. Papers devoted to the d-plane range question for S(Rn) are Gelfand-Gindikin and Graev [1982], Grinberg [1987], Richter [1986] and Gonzalez [1991]. This last paper gives the range as the kernel of a single 4th order differential operator on the space of d-planes. As shown by Gonzalez, the analog of Theorem 6.3 fails to hold for S(R"). An £ 2-version of Theorem 6.3 was given by Solmon [1976], p. 77. Proposition 6.13 was communicated to me by Schlichtkrull.

Some difficulties with the d-plane transform on L2 (Rn) are pointed out by Smith and Solmon [1975] and Solmon [1976], p. 68. In fact, the function f(x) = lxl-!n(log jxl)-1 (lxl ;::: 2), 0 otherwise, is square integrable on R" but is not integrable over any plane of dimension;::: ~·Nevertheless, see for example Rubin [1998a], Strichartz [1981] for £P-extensions of the d-plane transform.

§ 7. The applications to partial differential equations go in part back to Herglotz [1931]; see John [1955]. Other applications of the Radon transform to partial differential equations with constant coefficients can be found in Courant-Lax [1955], Gelfand-Shapiro [1955], John [1955], Borovikov [1959], Garding [1961] and Ludwig [1966]. Our discussion of the wave equation (Theorem 7.3 and Corollary 7.4) is closely related to the treatment in Lax-Phillips [1967], Ch. IV, where however, the dimension is assumed to be odd. Applications to general elliptic equations were given by John [1955].

While the Radon transform on Rn can be used to "reduce" partial differential equations to ordinary differential equations one can use a Radon type transform on a symmetric space X to ''reduce" an invariant differential operator D on X to a partial differential operator with constant coefficients. For an account of these applications see the author's monograph [1994b], Chapter V.

While the applications to differential equations are perhaps the most interesting to mathematicians, the tomographic applications of the X-ray transform have revolutionized medicine. These applications originated with Cormack [1963], [1964] and Hounsfield [1973]. For the approximate reconstruction problem, including Propositions 7.8 and 7.9 and refinements of Theorems 7.10, 7.11 see Smith, Solmon and Wagner [1977], Solmon [1976] and Hamaker and Solmon [1978]. Theorem 7.11 is due to Halperin [1962], the proof in the text to Amemiya and Ando [1965]. For an account of some of those applications see e.g. Deans [1983], Natterer [1986] and Ramm and Katsevich [1996]. Applications in radio astronomy appear in Bracewell and Riddle [1967].

CHAPTER II

A DUALITY IN INTEGRAL GEOMETRY. GENERALIZED RADON TRANSFORMS AND

ORBITAL INTEGRALS

§1 Homogeneous Spaces in Duality

The inversion formulas in Theorems 3.1, 3.5, 3.6 and 6.2, Ch. I suggest the general problem of determining a function on a manifold by means of its integrals over certain submanifolds. In order to provide a natural framework for such problems we consider the Radon transform f -+ 1 on lRn and its dual t.p-+ ((;from a group-theoretic point of view, motivated by the fact that the isometry group M(n) acts transitively both on lRn and on the hyperplane space pn. Thus

(1) lRn = M(n)JO(n), pn = M(n)JZ 2 X M(n -1),

where O(n) is the orthogonal group fixing the origin 0 E lRn and Z2 x M(n- 1) is the subgroup of M(n) leaving a certain hyperplane ~0 through 0 stable. (Z2 consists of the identity and the reflection in this hyperplane.)

We observe now that a point g1 O(n) in the first coset space above lies on a plane g2(Z2 x M(n- 1)) in the second if and only if these cosets, considered as subsets of M(n), have a point in common. In fact

91 · 0 C 92 · ~o ¢> 91 · 0 = 92h · 0 for some hE Z2 X M(n- 1) ¢> 91k = gzh for somek E O(n).

This leads to the following general setup. Let G be a locally compact group, X and 2 two left coset spaces of G,

(2) X=G/K, 2=G/H,

where K and Hare closed subgroups of G. Let L = K n H. We assume that the subset K H C G is closed. This is automatic if one of the groups K or H is compact.

Two elements x E X, ~ E 2 are said to be incident if as cosets in G they intersect. We put (see Fig. II.1)

x = {~ E 2 : x and ~incident} ~

~ = { x E X : x and ~ incident} .

54 Chapter II. A Duality in Integral Geometry.

X

FIGURE ILl.

Let x0 = { K} and ~0 = { H} denote the origins in X and 3, respectively. If II : G -+ G / H denotes the natural mapping then since x0 = K · ~0 we have

II-1(3- xo) = {g E G: gH f/. KH} = G- KH.

In particular II( G - K H) = 3 - x0 so since II is an open mapping, xo is a closed subset of 3. This proves

Lemma 1.1. Each x and each f is closed.

Using the notation Ag = gAg-1 (g E G, A C G) we have the following lemma.

Lemma 1.2. Let g,'Y E G, x E gK, ~ = '"fH. Then

x is an orbit of K 9 , f is an orbit of H'Y ,

and

Proof. By definition

(3) x = { 6H : 6H n gK =f. 0} = {gkH : k E K}

which is the orbit of the point gH under gK g-1 • The subgroup fixing gH is gKg-1 ngHg-1 = £9. Also (3) implies

x = g·xo f='Y·fo, where the dot · denotes the action of G on X and 3.

Lemma 1.3. Consider the subgroups

KH = {k E K: kHuk-1H C HK}

HK = {hEH: hKuh-1KcKH}.

The following properties are equivalent:

§1 Homogeneous Spaces in Duality 55

(a) KnH = KH = HK.

(b) The maps X -t X (X E X)> and e -t f ( e E 3) are injective.

We think of property (a) as a kind of transversality of K and H.

Proof. Suppose X1 = 91K, X2 = g2K and x1 = x2. Then by {3) 91 . Xo = Y1 ·xo so g·xo = xo if g = 91192· In particular g·eo c xo so g·eo = k·eo for some k E K. Hence k-1g =hE H so h·x0 = x0 , that is hK ·eo= K ·eo. As a relation in G, this means hK H = K H. In particular hK C K H. Since h·xo = x0 implies h-1 ·xo = x0 we have also h-1 K c KH so by {b) hE K which gives x1 = x2.

On the other hand, suppose the map x -t x is injective and suppose hE H satisfies h-1K U hK c KH. Then

hK ·eo c K ·eo and h-1 K ·eo c K ·eo.

By Lemma 1.2, h · x0 C x0 and h-1 · xo C xo. Thus h · x0 = xo whence by the assumption, h · x0 = x0 soh E K.

Thus we see that under the transversality assumption a) the elements e can be viewed as the subsets f of X and the elements x as the subsets x of 3. We say X and 3 are homogeneous spaces in duality.

The maps are also conveniently described by means of the following double fibration

(4) GfL

/~ GfK G/H

where p(gL) = gK, 1r(-yL) = -yH. In fact, by (3) we have

x = 1r(p-1 (x)) f = p(7r-1(e)).

We now prove some group-theoretic properties of the incidence, supplementing Lemma 1.3.

Theorem 1.4. (i} . We have the identification

G/L = {(x,e) EX X 2 :X and e incident}

via the bijection r : gL -t (gK, gH).

(ii} The property

KHK=G

is equivariant to the property:

Any two X1' X2 E X are incident to some e E 3. A similar statement holds for HKH =G.


(iii) The property

HKnKH=KUH

is equivalent to the property:

For any two x1 =I x2 in X there is at most one ~ E S incident to both. By symmetry, this is equivalent to the property:

For any ~1 =I 6 in S there is at most one x E X incident to both.

Proof. (i) The map is well-defined and injective. The surjectivity is clear because if 9Kn7H =10 then 9k = 7h and r(9kL) = (gK,7H).

(ii) We can take x2 = Xo. Writing x1 = 9K, ~ = 7H we have

xo, ~ incident <=? rh = k (some h E H, k E K) x1,~ incident <=? 7h1 = g1k1 (some h1 E H,k1 E K)

Thus if xo, x1 are incident to ~ we have 91 = kh-1h1k:;1. Conversely if 91 = k' h' k11 we put r = k' h' and then xo, x1 are incident to ~ = 1 H.

(iii) Suppose first KHnHK = KUH. Let x1 =I x2 in X. Suppose 6 =I ~2 inS are both incident to x1 and x2. Let Xi = giK, ~i = "{jH. Since Xi is incident to ~i there exist kij E K, hij E H such that

(5) i = 1, 2 j j = 1, 2.

Eliminating 9i and 'Yi we obtain

(6)

This being in K H n H K it lies in K U H. H the left hand side is in K then h2{hll E K so

contradicting x2 =I x1. Similarly if expression ( 6) is in H then k1 i k11 E H so by (5) we get the contradiction

Conversely, suppose K H n H K =I K U H. Then there exist h1 , h2, k1 , k2 such that h1k1 = k2h2 and h1k1 ~ K U H. Put x1 = h1K, ~2 = k2H. Then x1 =I xo, ~o =I 6, yet both ~o and 6 are incident to both xo and x1.

§2 The Radon Transform for the Double Fibration 57

Examples

(i) Points outside hyperplanes. We saw before that if in the coset space representation (1) O(n) is viewed as the isotropy group ofO and Z2M(n-1) is viewed as the isotropy group of a hyperplane through 0 then the abstract incidence notion is equivalent to the naive one: x E JR.n is incident to ~ E pn if and only if x E ~·

On the other hand we can also view Z2M(n- 1) as the isotropy group of a hyperplane ~o at a distance 8 > 0 from 0. (This amounts to a different embedding of the group Z2M(n-1) into M(n).) Then we have the following generalization.

Proposition 1.5. The point x E JR1' and the hyperplane ~ E pn are incident if and only if distance (x, ~) = 8.

Proof. Let x = gK, ~ = 1H where K = O(n), H = Z2M(n- 1). Then if gKn7H :f. 0, we have gk = "(h for some k E K, hE H. Now the orbit H ·0 consists of the two planes ~6 and ~~ parallel to ~o at a distance 8 from ~o. The relation

g. o ="(h. o E 'Y. <~.5 u ~n together with the fact that g and 1 are isometries shows that x has distance 8 from 1 · ~o = ~.

On the other hand if distance (x, ~) = 8 we have g·O E T(~6U~~) = 1H·O which means gK n 1H :f. 0.

(ii) Unit spheres. Let cro be a sphere in JR.n of radius one passing through the origin. Denoting by ~ the set of all unit spheres in JR.n we have the dual homogeneous spaces

(7) JR.n = M(n)IO(n); ~ = M(n)IO*(n)

where O*(n) is the set of rotations around the center of cr0 • Here a point x = gO(n) is incident to cr0 = 10*(n) if and only if x E cr.

§2 The Radon Transform for the Double Fibration

With K, H and L as in §I we assume now that K I L and HI L have positive measures dJ.Lo = dkL and dmo = dhL invariant under K and H, respectively. This is for example guaranteed if L is compact.

Lemma 2.1. Assume the transversality condition (a). Then there exists a measure on each x coinciding with dp.o on K I L = xo such that whenever g ·XI = x2 the measures on XI and x2 correspond under g. A similar statement holds for dm on f.


Proof. If x = 9 · x0 we transfer the measure dJ.to = dkL over on x by the map e -t 9 ·e. If 9 · Xo = 91 · xo then (9· xo)v = (91 ·xo)v so by Lemma 1.3, 9 · xo = 91 · xo so 9 = 91k with k E K. Since dJ.to is K-invariant the lemma follows.

The measures defined on each x and f under condition (a) are denoted by dJ.t and dm, respectively.

Definition. The Radon transform f -t f and its dual r.p -t :{; are defined by

(8) f(e) = fet(x) dm(x), c;5(x) = fx r.p(e) dJ.t(e).

whenever the integrals converge. Because of Lemma 1.1, this will always happen for f E Co(X), r.p E Co(2).

In the setup of Proposition 1.5, f{e) is the integral of f over the two hyperplanes at distance 0 from e and cp(x) is the average of r.p over the set of hyperplanes at distance o from x. For o = 0 we recover the transforms of Ch. I, §1.

Formula (8) can also be written in the group-theoretic terms,

(9) j('yH) = { f('yhK) dhL, jH/L

cp(9K) = { r.p(9kH) dkL . jK/L

Note that (9) serves as a definition even if condition (a) in Lemma 1.3 is not satisfied. In this abstract setup the spaces X and 2 have equal status. The theory in Ch. I, in particular Lemma 2.1, Theorems 2.4, 2.10, 3.1 thus raises the following problems:

Principal Problems:

A. Relate function spaces on X and on 2 by means of the transforms f --t j, r.p --t ip. In particular, determine their ranges and kernels.

B. Invert the transforms f -t j, r.p -t cp on suitable function spaces.

C. In the case when G is a Lie group so X and 2 are manifolds let D(X) and D(2) denote the algebras of G-invariant differential operators on X and 2<- respectively. Is there a map D -t D of D(X) into D(2) and a map E -t E of D(2) into D(X) such that

Although weaker assumptions would be sufficient, we assume now that the groups G, K, Hand L all have hi-invariant Haar measures d9, dk, dh

§2 The Radon Transform for the Double Fibration 59

and dl. These will then generate invariant measures dgK, dgH, dgL, dkL, dhL on GIK, GIH, GIL, KIL, HIL, respectively. This means that

(10)

and similarly dg and dh determine dgH, etc. Then

(11) { Q(gL) dgL = c { dgK { Q(gkL) dkL jG/L jG/K jK/L

for Q E Cc (GIL) where cis a constant. In fact, the integrals on both sides of (11) constitute invariant measures on GIL and thus must be proportional. However,

(12) fa F(g)dg = la/L ([ F(gl)d£) dgL

and

(13) [ F(k) dk = [/L ([ F(kl) d£) dkL.

Using (13) on (10) and combining with (11) we see that the constant c equals 1.

We shall now prove that f ~ f and r.p ~ if; are adjoint operators. We write dx for dgK and d~ for dgH.

Proposition 2.2. Let f E Cc(X), <p E Cc(S). Then j and if; are continuous and

Proof. The continuity statement is immediate from (9). We consider the function

P = (f o p)(<p orr)

on GIL. We integrate it over GIL in two ways using the double fibration (4). This amounts to using (11) and its analog with GIK replaced by GIH with Q = P. Since P(gkL) = f(gK)<p(gkH) the right hand side of (11) becomes

1 f(gK)ip(gK) dgK. G/K

If we treat G I H similarly, the lemma follows.


The result shows how to define the Radon transform and its dual for measures and, in case G is a Lie group, for distributions.

Definition. Let s be a measure on X of compact support. Its Radon transform is the functional son Cc(2) defined by

(14) S(cp) = s(<P).

Similarly a is defined by

(15) a(!) = u([) ' f E Cc(X)

if u is a compactly supported measure on 2.

Lemma 2.3. (i} If s is a compactly supported measure on X, s is a measure on 2. (ii} If s is a bounded measure on X and if Xo has finite measure then s as defined by (14} is a bounded measure.

Proof. (i) The measure s can be written as a difference s = s+ - s- of two positive measures, each of compact support. Then s = s+ - s is a difference of two positive functionals on Cc(2).

Since a positive functional is necessarily a measure, sis a measure. (ii) We have

sup J<P(x)J ~sup Jcp(.;)l Jlo(xo) X ~

so for a constant K,

IS(cp)J = Js(<P)J ~ KsupJ<Pl ~ K~to(xo)supJcpJ,

and the boundedness of s follows.

If G is a Lie group then (14), (15) with f E V(X), cp E V(2) serve to define the Radon transform s -+ s and the dual u -+ a for distributions s and a of compact support. We consider the spaces V(X) and £(X) (= C00 (X)) with their customary topologies (Chapter V,§l). The duals V'(X) and £'(X) then consist of the distributions on X and the distributions on X of compact support, respectively.

Proposition 2.4. The mappings

f E V(X) -+ f E £(2) cp E V(2) -+ <P E f(X)

are continuous. In particular,

s E t''(X) => s E V'(2)

u E £'(2) => a E V'(X).

§2 The Radon 'Transform for the Double Fibration 61

Proof. We have

(16) 1(g. eo) = ~ f(g. x) dmo(x). ho Let g run through a local cross section through e in G over a neighborhood of eo in 3. H (t1, ... , tn) are coordinates of g and (x1, ... ,xm) the coordinates of x E fa then (16) can be written in the form

F(t~, ... , tn) = J F(t1, ... ,tn; X1, ... 1 Xm) dx1 ... dxm.

Now it is clear that 1 E £(3) and that f --)- 1 is continuous, proving the proposition.

The result has the following refinement.

Proposition 2.5. Assume K compact. Then

(i) f --)-1 is a continuous mapping of1J(X) into 1J(3).

(ii) cp--)- if; is a continuous mapping of £(3) into £(X).

A self-contained proof is given in the author's book [1994b], Ch. I, § 3. The result has the following consequence.

Corollary 2.6. Assume K compact. Then t'' (Xf C £' (3), 1)' (3) v C 1)' (X).

In Chapter I we have given solutions to Problems A, B, C in some cases. Further examples will be given in § 4 of this chapter and Chapter III will include their solution for the antipodal manifolds for compact two-point homogeneous spaces.

The variety of the results for these examples make it doubtful that the individual results could be captured by a general theory. Our abstract setup in terms of homogeneous spaces in duality is therefore to be regarded as a framework for examples rather than as axioms for a general theory.

Nevertheless, certain general features emerge from the study of these examples. H dim X = dim 3 and f --)- 1 is injective the range consists of functions which are either arbitrary or at least subjected to rather weak conditions. As the difference dim 3 - dim X increases more conditions are imposed on the functions in the range. (See the example of the d-plane transform in lRn.) In the case when G is a Lie group there is a grouptheoretic explanation for this. Let D( G) denote the algebra of left-invariant differential operators on G. Since D(G) is generated by the left invariant vector fields on G, the action of G on X and on 3 induces homomorphisms

(17)

(18)

.X : D( G) ~ E(X) ,

A: D(G) ~ E(3),


where for a manifold M, E(M) denotes the algebra of all differential operators on M. Since f -t 1 and cp -t ij; commute with the action of G we have forD E D(G),

(19) (>..(D)f)v = A(D)1, (A(D)cp)v = >..(D)ij;.

Therefore A(D) annihilates the range off -t 1 if >..(D) = 0. In some cases, including the case of the d-plane transform in JR.n, the range is characterized as the null space of these operators A( D) (with >..(D) = 0).

§3 Orbital Integrals

As before let X = G / K be a homogeneous space with origin o = (K). Given x0 EX let Ga:o denote the subgroup of G leaving x0 fixed, i.e., the isotropy subgroup of Gat x0 •

Definition. A generalized sphere is an orbit Ga:0 • x in X of some point x E X under the isotropy subgroup at some point x0 E X.

Examples. (i) If X = JR.n, G = M(n) then the generalized spheres are just the spheres.

(ii) Let X be a locally compact subgroup L and G the product group L x L acting on Lon the right and left, the element (l1 ,l2 ) E L x L inducing action l -t l 1li21 on L. Let ~L denote the diagonal in L x L. If lo E L then the isotropy subgroup of l 0 is given by

(20)

and the orbit of l under it by

(LX L)t0 ·f.= lo(l01 l)L.

that is the left translate by l 0 of the conjugacy class of the element l 0 1i. Thus the generalized spheres in the group L are the left (or right) translates of its conjugacy classes.

Coming back to the general case X= G/K = G/Go we assume that Go, and therefore each Ga:0 , is unimodular. But Ga:0 ·X= Ga:0 /(Ga:0 )x so (Ga:0 )x unimodular implies the orbit Ga:0 • x has an invariant measure determined up to a constant factor. We can now consider the following general problem (following Problems A, B, C above).

D. Determine a function f on X in terms of its integrals over generalized spheres.

§4 Examples of Radon Transforms for Homogeneous Spaces in Duality 63

Remark 3.1. In this problem it is of course significant how the invariant measures on the various orbits are normalized.

(a) H Go is compact the problem above is rather trivial because each orbit Gx0 ·x has finite invariant measure so f(xo) is given as the limit as x -t xo of the average off over G.,0 • x.

(b) Suppose that for each xo EX there is a G.,0 -invariant open set Czo C X containing Xo in its closure such that for each x E C:~:0 the isotropy group (Gz0 )z is compact. The invariant measure on the orbit Gx0 ·X (xo E X,x E Cx0 ) can then be consistently normalized as follows: Fix a Haar measure dgo on Go. H Xo = g ·owe have Gx0 = gGog-1 and can carry dgo over to a measure dgz0 on Gx0 by means of the conjugation z -t gzg-1 (z EGo). Since dgo is hi-invariant, dgx 0 is independent of the choice of g satisfying Xo = g·o, and is hi-invariant. Since (Gzo)z is compact it has a unique Haar measure dgzo,z with total measure 1 and now dgz0 and dgz0 ,z determine canonically an invariant measure J..t on the orbit G.,0 • x = G.,0 j(Gz0 )z. We can therefore state Problem D in a more specific form.

D'. Express f(xo) in terms of integrals

{21)

For the case when X is an isotropic Lorentz manifold the assumptions above are satisfied (with C:~:0 consisting of the "timelike" rays from xo) and we shall obtain inCh. IV an explicit solution to Problem D' (Theorem 4.1, Ch. IV).

(c) H in Example (ii) above L is a semisimple Lie group Problem D is a basic step (Gelfand-Graev [1955], Barish-Chandra [1957]) in proving the Plancherel formula for the Fourier transform on L.

§4 Examples of Radon Transforms for Homogeneous Spaces in Duality

In this section we discuss some examples of the abstract formalism and problems set forth in the preceding sections §1-§2.

A. The Funk Transform.

This case goes back to Funk [1916] (preceding Radon's paper [1917]) where he proved that a symmetric function on S2 is determined by its great circle integrals. This is carried out in more detail and in greater generality in Chapter III, §1. Here we state the solution of Problem B for X = S2 , 3


0

FIGURE II.2. FIGURE II.3.

the set of all great circles, both as homogeneous spaces of 0(3). Given p ~ 0 let ~P E 3 have distance p from the North Pole o, Hp C 0(3) the subgroup leaving ~P invariant and K C 0(3) the subgroup fixing o. Then in the double fibration

0(3)/(K n Hp)

~~ X= 0(3)/K 3 = 0(3)/Hp

x E X and ~ E 3 are incident if and only if d(x, ~) = p. The proof is the same as that of Proposition 1.5. In order to invert the Funk transform f -+ 1 ( = fo) we invoke the transform <p -+ if;p. Note that (})~ (x) is the average of the integrals of f over the great circles ~ at distance p from x (see Figure II.2). As a special case of Theorem 1.11, Chapter III, we have the following inversion.

Theorem 4.1. The Funk transform f-+ 1 is (for f even) inverted by

(22) 1 { d 1" ~ v 2 2 _l } f(x) = -2 -d (f)cos-l(v)(x)v(u - v ) 2 dv . W U 0 u=l

Another inversion formula is

(23)

(Theorem 1.15, Chapter III), where Lis the Laplacian and S the integral operator given by (66)-(68), Chapter III. While (23) is short the operator S is only given in terms of a spherical harmonics expansion. Also Theorem 1.17, Ch. III shows that if f is even and if all its derivatives vanish on the equator then f vanishes outside the "arctic zones" C and C' if and only if](~) = 0 for all great circles ~disjoint from C and C' (Fig. II.3).


The Hyperbolic Plane H 2•

This remarkable object enters into several fields in mathematics. In particular, it offers at least two interesting cases of Radon transforms. We take H 2 as the disk D : jzj < 1 with the Riemannian structure

(24) ( ) (u,v) , dsz= jdzl2 u,v z = 2 (1 - jzj2) (1 - lzl2)2

if u and v are any tangent vectors at z E D. Here ( u, v) denotes the usual inner product on R2 • The Laplace-Beltrami operator for (24) is given by

( 2 2)2 ( 02 02 ) L= 1-x -y {)x2 + 8y2 .

The group G = SU(1, 1) of matrices

acts transitively on the unit disk by

(25) (~ ~) . z = ~z + b b a bz +a

and leaves the metric (24) invariant. The length of a curve 7(t) (a: st~ /3) is defined by

(26)

If 7(a:) = o, 7(/3) = x E R and 'Yo(t) = tx (0 ~ t s 1) then (26) shows easily that L('Y) ~ L('Y0 ) so 'Yo is a geodesic and the distanced satisfies

(27) -11 lxl _ 1 1 + lxl d( o, x) - 1 2 2 dt - 2 Iog -1 - 1- 1 . 0 -tx -x

Since G acts conformally on D the geodesics in H 2 are the circular arcs in lzl < 1 perpendicular to the boundary lzl = 1.

We consider now the following subgroups of G:

K = {ko= (~iB~-iB) :0~0<27r}

M {ko,k,.}, M 1 = {k0 ,k,.,k-~,k~}

A = (chtsht) {at= sht cht : t E lli},

N = { n = ( ~ + ix, -ix. ) z zx, 1- zx :X E R}

r = CSL(2, Z)C-1 ,


where C is the transformation w --+ ( w - i) I ( w + i) mapping the upper half-plane onto the unit disk.

The orbit A· o is the horizontal diameter and the orbits N ·(at· o) are the circles tangential to lzl = 1 at z = 1. Thus N A · o is the entire disk D so we see that G = NAK and also G = KAN.

B. The X-ray Transform in H 2 •

The (unoriented) geodesics for the metric (24) were mentioned above. Clearly the group G permutes these geodesics transitively (Fig. II.4). Let 3 be the set of all these geodesics. Let o denote the origin in H 2 and ~o the horizontal geodesic through o. Then

(28)

X = G I K' 3 = G I M 1 A.

We can also fix a geodesic ~P at distance p from o and write

(29)

X=GIK, 2=GIHp,

where Hp is the subgroup of G leaving ~P stable. Then for the homogeneous spaces (29), x and ~ are incident if and only if ~

FIGURE II.4.

geodesics inD

d(x, ~) = p. The transform f --+ f is inverted by means of the dual transform <p--+ lpp for (29). The inversion below is a special case of Theorem 1.10, Chapter III, and is the analog of (22). Note however the absence of v in the integrand. Observe also that the metric ds is renormalized by the factor 2 (so curvature is -1).

Theorem 4.2. The X-ray transform in H 2 with the metric

2 4ldzl2 ds = (1 - lzl2)2

is inverted by

(30)


Another inversion formula is

(31)

where S is the operator of convolution on H 2 with the function x -t cosh(d(x,o)) -1, (Theorem 1.14, Chapter III).

C. The Horocycles in H 2 •

Consider a family of geodesics with the same limit point on the boundary B. The horocycles in H 2 are by definition the orthogonal trajectories of such families of geodesics. Thus the horocycles are the circles tangential to lzl = 1 from the inside (Fig. II.5).

One such horocycle is ~o = N · o, the orbit of the origin o under the action of N. Since at · ~ is the horocycle with diameter (tanh t, 1) G acts transitively on the set 3 ofhorocycles. Now we take H 2 with the metric (24). Since G = K AN it is easy to see that M N is the subgroup leaving ~o invariant. Thus we have here

(32)

X=GIK, 3=GIMN. FIGURE II.5.

geodesics and horocycles inD

Furthermore each horocycle has the form ~ = kat · ~0 where kM E KIM and t E JR. are unique. Thus 3 ,....., KIM x A, which is also evident from the figure.

We observe now that the maps

'1/J:t--+at·O, cp:x-tn21 ·0

of JR. onto 'Yo and ~0 , respectively, are isometries. The first statement follows from (27) because

d(o, at) = d(o, tanh t) = t. For the second we note that

cp(x) = x(x + i)-1 , cp'(x) = i(x + i)-2


so

(rp' (x ), rp' (x ))<p(x) = (x2 + 1)-4 (1 - ix(x + i)-1 12)-2 = 1.

Thus we give A and N the Haar measures d(at) = dt and d(nx) = dx. Geometrically, the Radon transform on X relative to the horocycles is

defined by

(33) f(~) = h f(x) dm(x),

where dm is the measure on ~ induced by (24). Because of our remarks about rp, (33) becomes

(34) f(g · ~o) = [ f(gn · o) dn,

so the geometric definition (33) coincides with the group-theoretic one in (9). The dual transform is given by

(35) cp(g · o) = [ rp(gk · ~o) dk, (dk =dO /27r).

In order to invert the transform f ~ f we introduce the non-Euclidean analog of the operator A in Chapter I, §3. Let T be the distribution on lR given by

(36) Trp = ! L (sh t)-1rp(t) dt, rp E 'D(IR),

considered as the Cauchy principal value, and put T' = dT / dt. Let A be the operator on 'D(S) given by

(37) (Arp)(kat · ~o) = L rp(k~-s · ~o)e-s dT'(s).

Theorem 4.3. The Radon transform f ~ j for horocycles in H 2 is inverted by

(38)

We begin with a simple lemma.

Lemma 4.4. Let T be a distribution on R Then the operator r on 'D(S) given by the convolution

(rrp)(kat · ~o) = L rp(kat-s · ~o) dr(s)

is invariant under the action of G.


Proof. To understand the action of g E G on 2 ,...., (K/M) x A we write gk = k'at'd. Since each a E A normalizes N we have

Thus the action of g on 2 ~ (K / M) x A induces this fixed translation at -t at+t' on A. This translation commutes with the convolution by r so the lemma follows.

Since the operators A,~, v in (38) are all G-invariant it suffices to prove the formula at the origin o. We first consider the case when fisK-invariant, i.e., f(k · z) = f(z). Then by (34)

(39) f{at · ~o) = L f(atnz · o) dx.

Because of (27) we have

(40) izi =tanh d(o, z), cosh2 d(o, z) = (1 -lzl2)-2 •

Since

atnz · o = (sht- ixet)f(cht- ixet)

(40) shows that the distances= d(o, atnz · o) satisfies

(41)

Thus defining F on [1, oo) by

(42) F(ch2s) =!(tanh s),

we have

F'(ch2s) = f'(tanh s)(2shschs)-1

so, since f'(O) = 0, limu-tl F'(u) exists. The transform (39) now becomes (with xet = y)

(43)

We put


and invert this as follows:

f <p'(u + z2) dz = f F'(u + y2 + z2) dydz JR JR2

= 211" 100 F'(u+r2)rdr=7r 100 F'(u+p)dp,

so

-1rF(u) = L <p1(u+z2 )dz.

In particular,

so

f(o) = _.!. r <p1(1 + z2 ) dz = _.!. r <p1(ch2r)chrdr' 7r)R 7r)R

= _.!. r r F'(ch2t + y2 ) dychtdt 11" JR JR

f(o) = _ _!_ r !.!(et f(at. ~o))~. 211" JR dt sh t

Since ( et f) (at · ~0) is even ( cf. ( 43)) its derivative vanishes at t = 0 so the integral is well defined. With T as in (36), the last formula can be written

(44) 1 I t~ f(o) = -Tt (e !(at· ~o)),

11"

the prime indicating derivative. H f is not necessarily K -invariant we use (44) on the average

Jb(z) = i f(k · z) dk = 2~ 121r f(ke · z) d8.

Since fb(o) = f(o), (44) implies

(45) f(o) =.!. r [et(Jb)(at. ~o)] dT'(t). 11" JR

This can be written as the convolution at t = 0 of (Jbf<at · ~0) with the image of the distribution etTf. under t -+ -t. Since T' is even the right hand side of (45) is the convolution at t = 0 of fa. with e-tTf. Thus by (37)

f(o) = .!.(Afb)(~o). 11"

Since A and ~ commute with the K action this implies

f(o) =.!. f (Af)(k ·{o) = .!.(Af)v(o) 11" )K Tr

and this proves the theorem. Theorem 4.3 is of course the exact analog to Theorem 3.6 in Chapter I,

although we have not specified the decay conditions for f needed in generalizing Theorem 4.3.


D. The Poisson Integral as a Radon Transform.

Here we preserve the notation introduced for the hyperbolic plane H 2 . Now we consider the homogeneous spaces

(46) X=GJMAN, 3=GfK.

Then 3 is the disk D: lzl < 1. On the other hand, X is identified with the boundary B : lzl = 1, because when G acts on B, MAN is the subgroup fixing the point z = 1. Since G = KAN, each coset gMAN intersects eK. Thus each x E X is incident to each ~ E 3. Our abstract Radon transform (9) now takes the form

(47) j'(gK) = { f(gkMAN)dkL= { f(g·b)db, jK/L jB

= l f(b) d(g~~. b) db.

Writing g-1 in the form

we have

-1 (- z 1 ·e · g : ( -+ -z( + 1 ' g- . et = etrp'

. eie- z et'P - ----,-,-

- -zei8 + 1' dcp 1 -lzl2 d(} = !z-eiBI'

and this last expression is the classical Poisson kernel. Since gK = z, (47) becomes the classical Poisson integral

(48) ~ r 1-lzl2

f(z)= }Bf(b)lz-bl2 db.

Theorem 4.5. The Radon transform f-+ f for the homogeneous spaces {46) .is the classical Poisson integral {48). The inversion is given by the classical Schwarz theorem

(49) f(b) =lim f(z), f E C(B), z-tb

solving the Dirichlet problem for the disk.

We repeat the geometric proof of (49) from our booklet [1981] since it seems little known and is considerably shorter than the customary solution in textbooks of the Dirichlet problem for the disk. In (49) it suffices to consider the case b = 1. Because of ( 4 7),

~ 1 [2"' ·e f{tanh t) = f(at · 0) = 211" Jo f(at · e1 ) d(}

= _!__ [ 2"' ( ei8 +tanh t ) 211" }0 f tanh tei8 + 1 d(} ·


Letting t-+ +oo, (49) follows by the dominated convergence theorem.

The range question A for f -+ 1 is also answered by classical results for the Poisson integral; for example, the classical characterization of the Poisson integrals of bounded functions now takes the form

(50) L00 (B) = {rp E £ 00 (3) : Lrp = 0}.

The range characterization (50) is of course quite analogous to the range characterization for the X-ray transform described in Theorem 6.9, Chapter I. Both are realizations of the general expectations at the end of §2 that when dim X < dim 3 the range of the transform f -+ 1 should be given as the kernel of some differential operators. The analogy between (50) and Theorem 6.9 is even closer if we recall Gonzalez' theorem [1990b] that if we view the X-ray transform as a Radon transform between two homogeneous spaces of M(3) (see next example) then the range (83) in Theorem 6.9, Ch. I, can be described as the null space of a differential operator which is invariant under M(3). Furthermore, the dual transform rp-+ (/;maps £(3) on £(X). (See Corollary 4.7 below.)

Furthermore, John's mean value theorem for the X-ray transform (Corollary 6.12, Chapter I) now becomes the exact analog of Gauss' mean value theorem for harmonic functions.

What is the dual transform rp -+ (/; for the pair (46)? The invariant measure on MAN/ M = AN is the functional

(51) rp-+ r rp(an. o) dadn. JAN

The right hand side is just (/;(bo) where b0 = eM AN. If g = a1n1 the measure (51) is seen to be invariant under g. Thus it is a constant multiple of the surface element dz = (1-x2 -y2)-2 dxdy defined by (24). Since the maps t --+ at· o and x --+ n., · o were seen to be isometries, this constant factor is 1. Thus the measure (51) is invariant under each g E G. Writing rp9 (z) = rp(g · z) we know (rp9 )v = (/;9 so

'f;(g. bo) = r rpg(an) dadn = 'f;(bo). JAN

Thus the dual transform rp -+ (/; assigns to each rp E 'D(S) its integral over the disk.

Table II.1 summarizes the various results mentioned above about the Poisson integral and the X-ray transform. The inversion formulas and the ranges show subtle analogies as well as strong differences. The last item in the table comes from Corollary 4. 7 below for the case n = 3, d = 1.


Poisson Integral X-ray Transform

Coset X= SU(l, 1)/MAN X= M(3)/0(3) spaces 2 = SU(1, 1)/ K 2 = M(3)/(M(1) x 0(2))

f-tf f(z) = f8 f(b) f;J~I: db f(l) = fe f(p) dm(p)

r.p-tcp cp(x) = fs r.p(~) ~ ip(x) =average of r.p over set of l through x

Inversion f(b) = limz-tb f(z) f = ~(-L)l/2((J)v)

Range of L00(Xf = V(Xf= f-tf {r.p E L00 (S): Lr.p = 0} {r.p E V(S): A(l~ -7]j-1cp) = 0}

Range Gauss' mean Mean value property for characteri- value theorem hyperboloids of revolution zation

Range of e(S)v = C e(s)v = e(X)

r.p-tcp

TABLE ILl. Analogies between the Poisson Integral and the X-ray Transform.

E. The d-plane Transform.

We now review briefly the d-plane transform from a group theoretic standpoint. As in (1) we write

(52)

X= Rn = M(n)fO(n), 2 = G(d,n) = M(n)f(M(d) x O(n- d)),

where M(d) xO(n-d) is the subgroup ofM(n) preserving a certain d-plane ~o through the origin. Since the homogeneous spaces

O(n)fO(n) n (M(d) x O(n- d)) = O(n)f(O(d) x O(n- d))

and

(M(d) X O(n- d))/O(n) n (M(d) X O(n- d))= M(d)/O(d)

have unique invariant measures the group-theoretic transforms (9) reduce to the transforms (52), (53) in Chapter I. The range of the d-plane transform is described by Theorem 6.3 and the equivalent Theorem 6.5 in Chapter I. It was shown by Richter [1986a] that the differential operators in Theorem 6.5 could be replaced by M(n)-induced second order differential


operators and then Gonzalez [1990b] showed that the whole system could be replaced by a single fourth order M{n)-invariant differential operator onS.

Writing {52) for simplicity in the form

{53) X=G/K, S=G/H

we shall now discuss the range question for the dual transform <p -+ ij; by invoking the d-plane transform on £'(X).

Theorem 4.6. Let N denote the kernel of the dual transform on £{3). Then the range of S -+ S on £'(X) is given· by

£'(X)= {:E E £'(S) : :E(N) = 0}.

The inclusion C is clear from the definitions (14),{15) and Proposition 2.5. The converse is proved by the author in (1983aJ and [1994b], Ch. I, §2 for d = n - 1; the proof is also valid for general d.

For Frechet spaces E and F one has the following classical result. A continuous mapping a: E-+ F is surjective if the transpose ta: F' -+ E' is injective and has a closed image. Taking E = £(3), F = £{X), a as the dual transform <p -+ ij;, the transpose ta is the Radon transform on £'(X). By Theorem 4.6, ta does have a closed image and by Theorem 5.5, Ch. I (extended to any d) ta is injective. Thus we have the following result (Hertle [1984] for d = n - 1) expressing the surjectivity of a.

Corollary 4.7. Every f E £(Rn) is the dual transform f = ij; of a smooth d-plane function <p.

F. Grassmann Manifolds.

We consider now the (affine) Grassmann manifolds G(p,n) and G(q,n) where p + q = n - 1. If p = 0 we have the original case of points and hyperplanes. Both are homogeneous spaces of the group M(n) and we represent them accordingly as coset spaces

(54) X= M(n)fHv, S = M(n)/Hq.

Here we take Hp as the isotropy group of a p-plane x0 through the origin 0 ERn, Hq as the isotropy group of a q-plane eo through 0, perpendicular to xo. Then

Hp ,..., M(p) x O(n- p), Hq = M(q) x O(n- q).

Also

§4 Examples of Radon 'J.ransforms for Homogeneous Spaces in Duality 75

the set of p-planes intersecting eo orthogonally. It is then easy to see that

X is incident to e {:} X J_ e 1 X n e #: 0 .

Consider as in Chapter I, §6 the mapping

1r : G(p,n)-+ G9 ,n

given by parallel translating a p-plane to one such through the origin. If u E G9 ,n, the fiber F = 1r-1 (u) is naturally identified with the Euclidean space u.L. Consider the linear operator 0 9 on &(G(p,n)) given by

(55)

Here Lp is the Laplacian on F and bar denotes restriction. Then one can prove that 0 9 is a differential operator on G(p, n) which is invariant under

the action of M(n). Let f -+ j, cp -+ ij; be the Radon transform and its

dual corresponding to the pair (54). Then iCe) represents the integral of f over all p-planes X intersecting e under a right angle. For n odd this is inverted as follows (Gonzalez [1984, 1987]).

Theorem 4.8. Let p,q E z+ such that p + q + 1 = n is odd. Then the transform f-+ j from G(p,n) to G(q,n) is inverted by the formula

C9 ,qJ = ((Oq)(n-l)/2 j)v, f E V(G(p,n))

where C9 ,q is a constant.

If p = 0 this reduces to Theorem 3.6, Ch. I.

G. Half-lines in a Half-plane.

In this example X denotes the half-plane {(a, b) E R2 : a> 0} viewed as a subset ofthe plane {(a, b, 1) E JR3 }. The group G of matrices

(a 0 0)

(a,{j,"f) = {j 1 'Y E GL(3,R), 0 0 1

a>O

acts transitively on X with the action

(a, {3, 'Y) 0 (a, b) = (aa, {ja + b + 'Y).

This is the restriction of the action of GL(3, JR.) on R3 • The isotropy group of the point xo = (1, 0) is the group

K = {(1, {j - {j) : {j E JR.} •


Let 2 denote the set of half-lines in X which end on the boundary 8X = 0 x R These lines are given by

ev,w = {(t,v + tw) : t > 0}

for arbitrary v, wE R Thus 2 can be identified with JR. x R The action of G on X induces a transitive action of G on S which is given by

w+/3 (a,f3,'Y)O(v,w) = (v+'Y,--).

a

(Here we have for simplicity written (v, w) instead of ev,w·) The isotropy group of the point eo = (0, 0) (the x-axis) is

H = {(a,O,O) :a> 0} =JR.~,

the multiplicative group of the positive real numbers. Thus we have the identifications

(56) X=G/K, S=G/H.

The group K n H is now trivial so the Radon transform and its dual for the double fibration in (56) are defined by

(57)

(58)

f(gH) = i f(ghK) dh,

ip(gK) = x(g) [ cp(gkH) dk,

where xis the homomorphism (a,/3 ,'Y)-+ a-1 of G onto~- The reason for the presence of xis that we wish Proposition 2.2 to remain valid even if G is not unimodular. In (57) and (58) we have the Haar measures

(59) dk(1,{3-/3) = df3, dh(a,O,O) = dafa.

Also, if g = (a, {3, 'Y), h = (a, 0, 0), k = (1, b, -b) then

gH = ('Y,{3ja),

gK=(a,f3+'Y),

ghK= (aa,{3a+'Y)

gkH = (-b+'Y, ~)

so (57)-(58) become

f('Y, {3ja) = f f(aa,f3a + 'Y) da At+ a

ip(a,/3 + 'Y) = a-1 k <p(-b+'Y,~)db.


Changing variables these can be written

(60) f(v,w) = r f (a, v + aw) da ' JR.+ a

(61) if;( a, b) = lcp(b-as,s)ds a>O.

Note that in (60) the integration takes place over all points on the line ev,w and in {61) the integration takes place over the set of lines eb-as,s all of which pass through the point (a, b). This is an a posteriori verification of the fact that our incidence for the pair (56) amounts to X E e.

From (60)-{61) we see that f -+ f, cp -+ if; are adjoint relative to the measures daa db and dv dw:

(62) r { f(a,b)<f;(a,b)da db= { { f(v,w)cp(v,w)dvdw. JR. JR.~ a JR. JR.

The proof is a routine computation. We recall (Chapter V) that (-L )112 is defined on the space of rapidly

decreasing functions on IR by

(63) ((-£) 1127/J)~ (r) = lrl~(r)

and we define A on S(S)(= S(JR2 )) by having (-£)112 only act on the second variable:

(64) (Acp)(v,w) = ((-L) 112cp(v, ·))(w).

Viewing (-£)112 as the Riesz potential J-1 on IR (Chapter V, §5) it is easy to see that if cpc(v,w) = cp(v, 7) then

(65)

The Radon transform (57) is now inverted by the following theorem.

Theorem 4.9. Let f E 1J(X). Then

Proof. In order to use the Fourier transform F -+ F on JR2 and on IR we need functions defined on all of JR2 • Thus we define

{ !. f (!. -b) a > 0 ,

f*(a,b) = a ao' a a:$ 0.


Then

f(a,b) = !r (!, -~) a a a

= a-1(211')-2 I I J*(e,.,)ei<!;-~> cie d.,

= (211')-2 I I J*(ae +h.,, "l)eie cie d.,

= a(211')-2 11 1e1r«a + ah.,)e, a.,e)eie d{ d.,.

Next we express the Fourier transform in terms of the Radon transform. We have

f*((a + ah.,)e, a11e) = I I f*(x, y)e-iz(a+abfl)ee-iya,e dx dy

= f f ! f (!, _!) e-iz(a+ab11)ee-iyafle dxdy JJR.lz'?:,O X X X

This last expression is

where"' denotes the !-dimensional Fourier transform (in the second variable). Thus

f(a, b)= a(211')-2 I I lel<ft"(b + .,-1, -a.,e)eie de d.,.

However F(ce) = lci-1 (Fc)~(e) so by (65)

f(a,b) = a(211')-2 I I lel((f)afj)~(b + .,-1, -e)eie ciela11l-1 d11

= (211')-1 I A((f)a1J)(b + .,-1, -1)1"11-1 d11

= (211')-1 I la"11-1(Ai)a11 (b + .,-1, -1)1"11-1 d11

= a-1(211')-1 I (Ai)(b + .,-1' -(a"l)-1 )"1-2 d"l

so

/(a, b) = (211')-1 L<Af)(b-av,v)dv

= (211')-1(Aj)v(a,b).


proving the theorem.

Remark 4.10. It is of interest to compare this theorem with Theorem 3.6,

Ch. I. If f E V( X) is extended to all of JR2 by defining it 0 in the left

half plane then Theorem 3.6 does give a formula expressing fin terms of

its integrals over half-lines in a strikingly similar fashion. Note however

that while the operators f -+ 1, cp -+ cp are in the two cases defined by

integration over the same sets (points on a half-line, half-lines through a

point) the measures in the two cases are different. Thus it is remarkable

that the inversion formulas look exactly the same.

H. Theta Series and Cusp Forms.

Let G denote the group SL(2, JR) of 2 x 2 matrices of determinant one and

r the modular group SL(2, Z). Let N denote the unipotent group ( ~ ~ )

where n E lR and consider the homogeneous spaces

(66) X=GfN, S=Gjr.

Under the usual action of G on JR2 , N is the isotropy subgroup of (1, 0) so

X can be identified with JR2 - (0), whereas Sis of course 3-dimensional.

In number theory one is interested in decomposing the space £ 2 ( G jr)

into G-invariant irreducible subspaces. We now give a rough description of

this by means of the transforms f-+ f and cp-+ ip. As customary we put roo= rnN; our transforms (9) then take the form

f(gr) = L !(91N), f/foo

ip(gN) = { cp(gnr) dnr""' . }N/foo

Since N jr 00 is the circle group, cp(gN) is just the constant term in the

Fourier expansion of the function nr oo -+ cp(gnr). The null space L~( G jr)

in £ 2 ( G jr) of the operator cp -+ ip is called the space of cusp forms and the

series for j is called theta series. According to Prop. 2.2 they constitute

the orthogonal complement of the image Cc(Xf We have now the G-invariant decomposition

(67)

where (- denoting closure)

(68)


and as mentioned above,

(69)

It is known (cf. Selberg [1962], Godement (1966]) that the representation of G on L~(Gjr) is the continuous direct sum of the irreducible representations of G from the principal series whereas the representation of G on L~(Gjr) is the discrete direct sum of irreducible representations each occurring with finite multiplicity.

In conclusion we note that the determination of a function in JRn in terms of its integrals over unit spheres (John [1955]) can be regarded as a solution to the first half of Problem Bin §2 for the double fibration (4).


The Radon transform and its dual for a double fibration

(1) Z=Gf(KnH)

~~ X=G/K 'B=G/H

was introduced in the author's paper [1966a]. The results of §1-§2 are from there and from [1994b]. The definition uses the concept of incidence for X = G / K and S = G/H which goes back to Chern [1942]. Even when the elements of Scan be viewed as subsets of X and vice versa (Lemma 1.3) it can be essential for the inversion off-+ f not to restrict the incidence to the naive one x E {. (See for example the classical case X = S2 , S = set of great circles where in Theorem 4.1 a more general incidence is essential.) The double fibration in (1) was generalized in Gelfand, Graev and Shapiro [1969], by relaxing the homogeneity assumption.

For the case of geodesics in constant curvature spaces (Examples A, B in §4) see notes to Ch. III.

The proof of Theorem 4.3 (a special case of the author's inversion formula in [1964], [1965b]) makes use of a method by Godement [1957] in another context. Another version of the inversion (38) for H 2 (and Hn) is given in Gelfand-GraevVilenkin [1966]. A further inversion of the horocycle transform in H 2 (and Hn), somewhat analogous to (30) for the X-ray transform, is given by Berenstein and Tarabusi [1994].

The analogy suggested above between the X-ray transform and the horocycle transform in H 2 goes even further in H 3 • There the 2-dimensional transform for totally geodesic submanifolds has the same inversion formula as the horocycle transform (Helgason [1994b], p. 209).

For a treatment of the horocycle transform on a Riemannian symmetric space see the author's monograph [1994b], Chapter II, where Problems A, B, C in §2 are discussed in detail along with some applications to differential equations and


group representations. See also Quinto [1993a] and Gonzalez and Quinto [1994] for new proofs of the support theorem.

Example G is from Hilgert's paper [1994], where a related Fourier transform theory is also established. It has a formal analogy to the Fourier analysis on H 2

developed by the author in [1965b] and [1972].

CHAPTER III

THE RADON TRANSFORM ON TWO-POINT HOMOGENEOUS SPACES

Let X be a complete Riemannian manifold, x a point in X and X:c the tangent space to X at x. Let Expz denote the mapping of X:c into X given by Expz(u) = 'Yu(1) where t-+ 'Yu(t) is the geodesic in X through x with tangent vector u at x = 'Yu(O).

A connected submanifold S of a Riemannian manifold X is said to be totally geodesic if each geodesic in X which is tangential to S at a point lies entirely in S.

The totally geodesic submanifolds of Rn are the planes in JRn. Therefore, in generalizing the Radon transform to Riemannian manifolds, it is natural to consider integration over totally geodesic submanifolds. In order to have enough totally geodesic submanifolds at our disposal we consider in this section Riemannian manifolds X which are two-point homogeneous in the sense that for any two-point pairs p, q E X p1 , q1 E X, satisfying d(p, q) = d(p1 , q1), (where d = distance), there exists an isometry g of X such that g · p = rf, g · q = q1• We start with the subclass of Riemannian manifolds with the richest supply of totally geodesic submanifolds, namely the spaces of constant curvature.

While §1, which constitutes most of this chapter, is elementary, §2-§5 will involve a bit of Lie group theory.

§1 Spaces of Constant Curvature. Inversion and Support Theorems

Let X be a simply connected complete Riemannian manifold of dimension n ~ 2 and constant sectional curvature.

Lemma 1.1. Let x E X, V a subspace of the tangent space Xz. Then Exp., (V) is a totally geodesic submanifold of X.

Proof. For this we choose a specific embedding of X into JRn+l , and assume for simplicity the curvature is e(= ±1). Consider the quadratic form

Be(x) = x~ + · · · + x; + ex;+l

and the quadric Qe given by Be(x) = e. The orthogonal group O(Be) acts transitively on Qe. The form Be is positive definite on the tangent space JRn x (0) to Qe at x0 = (0, ... , 0, 1); by the transitivity Be induces a positive definite quadratic form at each point of Qe, turning Qe into a

84 Chapter III. The Radon Transform on Two-point Homogeneous Spaces

Riemannian manifold, on which O(Be) acts as a transitive group of isometries. The isotropy subgroup at the point x0 is isomorphic to O(n) and its acts transitively on the set of 2-dimensional subspaces of the tangent space (Qe)xo. It follows that all sectional curvatures at x0 are the same, namely €, so by homogeneity, Q, has constant curvature €. In order to work with connected manifolds, we replace Q_1 by its intersection Q~1 with the half-space Xn+I > 0. Then Q+I and Q~1 are simply connected complete Riemannian manifolds of constant curvature. Since such manic. folds are uniquely determined by the dimension and the curvature it follows that we can identify X with Q+I or Q~1 .

The geodesic in X through x0 with tangent vector (1, 0, ... , 0) will be left point-wise fixed by the isometry

This geodesic is therefore the intersection of X with the two-plane x2 = · · · = Xn = 0 in JRn+l . By the transitivity of 0 ( n) all geodesics in X through x0 are intersections of X with two-planes through 0. By the transitivity of O(Q,) it then follows that the geodesics in X are precisely the nonempty intersections of X with two-planes through the origin.

Now if V C Xxo is a subspace, Expzo (V) is by the above the intersection of X with the subspace of JRn+l spanned by V and x0 • Thus Expxo (V) is a quadric in V + IRx0 and its Riemannian structure induced by X is the same as induced by the restriction B,I(V + IRx0 ). Thus, by the above, the geodesics in Expxo (V) are obtained by intersecting it with two-planes in V + IRx0 through 0. Consequently, the geodesics in Expzo (V) are geodesics in X so Expxo (V) is a totally geodesic submanifold of X. By the homogeneity of X this holds with x0 replaced by an arbitrary point x E X. The lemma is proved.

In accordance with the viewpoint of Ch. II we consider X as a homogeneous space of the identity component G of the group O(Q,). Let K denote the isotropy subgroup of G at the point x0 = (0, ... , 0, 1). Then K can be identified with the special orthogonal group SO(n). Let k be a fixed integer, 1 :::; k:::; n- 1; let eo c X be a fixed totally geodesic submanifold of dimension k passing through x0 and let H be the subgroup of G leaving eo invariant. We have then

(I) X=G/K, '2=0/H,

3 denoting the set of totally geodesic k-dimensional submanifolds of X. Since x0 E eo it is clear that the abstract incidence notion boils down to the naive one, in other words: The cosets X= gK e = "(H have a point in common if and only if X E e. In fact

§1 Spaces of Constant Curvature. Inversion and Support Theorems 85

A. The Hyperbolic Space

We take first the case of negative curvature, that is € = -1. The transform f --1 f is now given by

(2) f(e) = l f(x) dm(x)

~ being any k-dimensional totally geodesic submanifold of X (1 ::::; k ::::; n-1) with the induced Riemannian structure and dm the corresponding measure. From our description of the geodesics in X it is clear that any two points in X can be joined by a unique geodesic. Let d be a distance function on X, and for simplicity we write o for the origin x 0 in X. Consider now geodesic polar-coordinates for X at o; this is a mapping

Exp0 Y --1 (r,fJt, ... ,Bn-1),

where Y runs through the tangent space X 0 , r = IYI (the norm given by the Riemannian structure) and (81, ... , Bn-1) are coordinates of the unit vector Y/IYI· Then the Riemannian structure of X is given by

(3)

where da2 is the Riemannian structure

n-1 L Yii(el> ···,en-d dei dei

i,j=l

on the unit sphere in X 0 • The surface area A(r) and volume V(r) = J; A(t) dt of a sphere in X of radius rare thus given by

(4) A(r) = On(sinh r)n-1 , V(r) =On 1r sinhn-1 tdt

so V(r) increases like e<n-1)r. This explains the growth condition in the next result where d(o, e) denotes the distance of 0 to the manifold e. Theorem 1.2. (The support theorem.) Suppose f E C(X) satisfies

(i) For each integer m > 0, f(x)emd(o,x) is bounded.

(ii) There exists a number R > 0 such that

f(~) = 0 for d(o,~) > R.

Then

f(x)=O ford(o,x)>R.


Taking R -t 0 we obtain the following consequence.

Corollary 1.3. The Radon transform f -t 1 is one-to-one on the space of continuous functions on X satisfying condition {i) of "exponential decrease".

Proof of Theorem 1.2. Using smoothing of the form

l cp(g)f(g-1 · x) dg

(cp E 'D(G), dg Haar measure on G) we can (as in Theorem 2.6, Ch. I) assume that f E £(X).

We first consider the case when f in (2) is a radial function. Let P denote the point in ~ at the minimum distance p = d( o, ~) from o, let Q E ~ be arbitrary and let

q = d(o,Q),

Since ~ is totally geodesic d(P, Q) is also the diStance between P and Q in ~. Consider now the totally geodesic plane 1r through the geodesics oP and oQ as given by Lemma 1.1 (Fig. III.!). Since a totally geodesic submanifold contains the geodesic joining any two of its points, 1r contains the geodesic PQ. The angle oPQ being 90° (see e.g. [DS], p. 77) we conclude by hyperbolic trigonometry, (see e.g. Coxeter (1957])

r = d(P,Q).

(5) cosh q = cosh p cosh r .

FIGURE III.l .

Since f is radial it follows from (5) that the restriction /I~ is constant on spheres in ~ with center P. Since these have area nk(sinh r)k-1 formula (2) takes the form

(6) 1(~) = nk 100 f(Q)(sinh r)k-1 dr.

Since f is a radial function it is invariant under the subgroup K c G which fixes o. But K is not only transitive on each sphere Sr ( o) with center o, it is for each fixed k transitive on the set of k-dimensional totally geodesic


submanifolds which are tangent to Sr(o). Consequently, f{e) depends only on the distance d(o, e). Thus we can write

f(Q) = F{cosh q), f(e) = F(cosh p)

for certain !-variable functions F and F, so by (5) we obtain

{7) F(coshp) = nk 100 F(coshp cosh r)(sinh r)k-1 dr.

Writing here t =cosh p, s =cosh r this reduces to

(8) F(t) = nk 100 F(ts)(s2 - l)(k-2)12 ds.

Here we substitute u = (ts)-1 and then put v = r 1. Then (8) becomes

v-1 F(v-1) = nk LIJ {F(u-1)u-k}(v2- u2)(k-2)/2 du.

This integral equation is of the form (19), Ch. I so we get the following analog of (20), Ch. I:

(9) F(u-1)u-k = cu (-d-) k t' (u2- v2)(k-2)/2 F(v-1) dv d(u2 ) lo '

where cis a constant. Now by assumption (ii) F(cosh p) = 0 if p > R. Thus

F(v-1) = 0 if 0 < v <(cosh R)-1 .

From (9) we can then conclude

F(u-1) = 0 if u <(cosh R)-1

which means f ( x) = 0 for d( o, x) > R. This proves the theorem for f radial. Next we consider an arbitrary f E E(X) satisfying (i), (ii) . Fix x E X

and if dk is the normalized Haar measure on K consider the integral

Fz(Y)= [f(gk·y)dk, yEX,

where g E G is an element such that g · o = x. Clearly, F:z: (y) is the average of f on the sphere with center x, passing through g · y. The function F:z: satisfies the decay condition (i) and it is radial. Moreover,

(10)


We now need the following estimate

(11) d(o, gk · ~) 2:: d{o, ~) - d(o,g · o).

For this let x 0 be a point on~ closest to k-1g-1 · o. Then by the triangle inequality

d(o,gk·~)=d(k-1g-1 ·o,~) > d(o,x0 )-d(o,k-1g-1 ·o)

> d( o, ~) - d( o, g . 0) .

Thus it follows by (ii) that

Fx(~) = 0 if d(o,~) > d(o,x) +R.

Since Fx is radial this implies by the first part of the proof that

(12) [f(gk·y)dk=O

if

(13) d(o,y) > d(o,g · o) + R.

But the set {gk · y : k E K} is the sphere Sd(o,y) (g · o) with center g · o and radius d(o,y); furthermore, the inequality in (13) implies the inclusion relation

(14)

for the balls. But considering the part in B R ( o) of the geodesic through o and g ·owe see that conversely relation (14) implies (13). Theorem 1.2 will therefore be proved if we establish the following lemma.

Lemma 1.4. Let f E C(X) satisfy the conditions:

{i) For each integer m > 0, f(x)emd(o,x) is bounded.

{ii) There exists a number R > 0 such that the surface integral

Is f(s) dw(s) = 0,

whenever the spheres S encloses the ball BR(o).

Then

f(x) = 0 for d(o,x) > R.


Proof. This lemma is the exact analog of Lemma 2.7, Ch. I, whose proof, however, used the vector space structure of JRn. By using a special model of the hyperbolic space we shall nevertheless adapt the proof to the present situation. As before we may assume f is smooth, i.e., f E £(X).

Consider the unit ball { x E JRn : I:~ xt < 1} with the Riemannian structure

(15)

where

This Riemannian manifold is well known to have constant curvature -1 so we can use it for a model of X. This model is useful here because the spheres in X are the ordinary Euclidean spheres inside the ball. This fact is obvious for the spheres E with center 0. For the general statement it suffices to prove that if T is the geodesic symmetry with respect to a point (which we can take on the x1-axis) then T(E) is a Euclidean sphere. The unit disk D in the x1x2-plane is totally geodesic in X, hence invariant under T. Now the isometries of the non-Euclidean disk D are generated by the complex conjugation x1 + ix2 ---+ x1 - ix2 and fractional linear transformations so they map Euclidean circles into Euclidean circles. In particular T(E n D) = T(E) n D is a Euclidean circle. But T commutes with the rotations around the x1-axis. Thus T(E) is invariant under such rotations and intersects D in a circle; hence it is a Euclidean sphere.

After these preliminaries we pass to the proof of Lemma 1.4. Let S = Sr (y) be a sphere in X enclosing Br ( o) and let Br (y) denote the corresponding ball. Expressing the exterior X- Br(Y) as a union of spheres in X with center y we deduce from assumption (ii)

(16) f j(x) dx = f f(x) dx, jB~(Y) lx

which is a constant for small variations in r and y. The Riemannian measure dx is given by

(17)

where dx0 = dx1 ... dxn is the Euclidean volume element. Let ro and y0 ,

respectively, denote the Euclidean radius and Euclidean center of Sr(y). Then Sr.(Yo) = Sr(y),Br.(Yo) = Br(Y) set-theoretically and by (16) and (17)

(18) 1 f(xo)p(xot dxo = const., B~0 (yo)


for small variations in r0 and y0 ; thus by differentiation with respect to r0 ,

(19) [ f(so)p(so)n dwo(so) = 0, 1 S~0 (Yo)

where dw0 is the Euclidean surface element. Putting f*(x) = f(x)p(x)n we have by (18)

so by differentiating with respect to Yo, we get

[ (od*)(yo + Xo) dxo = 0. 1 B~0 (o) Using the divergence theorem (26), Chapter I, §2, on the vector field F(x0 ) = f*(y0 +xo)Oi defined in a neighborhood of Bro (0) the last equation implies

[ f*(Yo + s)si dwo(s) = 0 1 8~0 (0)

which in combination with (19) gives

(20)

The Euclidean and the non-Euclidean Riemannian structures on SrJYo) differ by the factor p2 • It follows that dw = p(s)n-l dw0 so (20) takes the form

(21) [ f(s)p(s)si dw(s) = 0. 1 s~(y)

We have thus proved that the function x --* f(x)p(x)xi satisfies the assumptions of the theorem. By iteration we obtain

(22) [ f(s)p(s)ksi 1 ... si,. dw(s) = 0. 1 S~(y) In particular, this holds with y = 0 and r > R. Then p( s) = constant and (22) gives f = 0 outside BR(o) by the Weierstrass approximation theorem. Now Theorem 1.2 is proved.

Now let L denote the Laplace-Beltrami operator on X. (See Ch. IV, §1 for the definition.) Because of formula (3) for the Riemannian structure of X, Lis given by

(23) a2 a

L = or2 + (n- 1) coth r or +(sinh r)-2 Ls


where Ls is the Laplace-Beltrami operator on the unit sphere in X0 . We consider also for each r ~ 0 the mean value operator Mr defined by

(Mrf)(x) = A(1 ) { f(s)dw(s). r J S~(x)

As we saw before this can also be written

(24) (Mr f)(g · o) = [ f(gk · y) dk

if g E G is arbitrary and y E X is such that r = d( o, y). If f is an analytic function one can, by expanding it in a Taylor series, prove from (24) that Mr is a certain power series in L (cf. Helgason [1959], pp. 270-272). In particular we have the commutativity

(25)

This in turn implies the "Darboux equation"

(26) Lx(F(x,y)) = Ly(F(x,y))

for the function F(x, y) = (Md(o,y) f)(x). In fact, using (24) and (25) we have if g · o = x, r = d( o, y)

Lx(F(x, y)) = (LMr f)(x) = (Mr Lf)(x)

= [ (Lf)(gk · y) dk = [ (Ly(f(gk · y))) dk

the last equation following from the invariance of the Laplacian under the isometry gk. But this last expression is Ly(F(x, y)).

We remark that the analog of Lemma 2.13 in Ch. IV which also holds here would give another proof of (25) and (26).

For a fixed integer k(1 ~ k ~ n - 1) let 3 denote the manifold of all k-dimensional totally geodesic submanifolds of X. If <p is a continuous function on 3 we denote by ip the point function

ip(x) = f <p(~) dp,(~), l:vEf.

where p, is the unique measure on the (compact) space of~ passing through x, invariant under all rotations around x and having total measure one.

Theorem 1.5. (The inversion formula.} For k even let Qk denote the polynomial

Qk(z)=[z + (k-1)(n-k)][z + (k-3)(n-k+2)] ... [z+1· (n-2)]


of degree k/2. The k-dimensional Radon transform on X is then inverted by the formula

Here c is the constant

(27) c = ( -41r)k/2r(n/2)/f((n- k)/2).

The formula holds also iff satisfies the decay condition {i) in Corollary 4.1.

Proof. Fix ~ E 3 passing through the origin o E X. H x E X fix g E G such that g · o = x. Ask runs through K, gk ·~runs through the set of totally geodesic submanifolds of X passing through x and

ij;(g · o) = [ rp(gk · ~) dk.

Hence

where r = d( o, y). But since ~ is totally geodesic in X, it has also constant curvature -1 and two points in ~ have the same distance in ~ as in X. Thus we have

(28)

We apply L to both sides and use (23). Then

where Lr is the "radial part" ~ + (n- 1) coth r tr of L. Putting now F(r) = (Mr f)(x) we have the following result.

Lemma 1.6. Let m be an integer 0 < m < n = dim X. Then

100 sinhmrLrFdr =

(m+1-n)[m 100 sinhmrF(r)dr+(m-1) 100 sinhm-2 rF(r)dr].

Ifm = 1 the term (m -1) J000 sinhm-2 rF(r) dr should be replaced by F(O).


This follows by repeated integration by parts. From this lemma combined with the Darboux equation (26) in the form

(30)

we deduce

[La:+ m(n- m- 1)] 100 sinhm r(Mr f)(x) dr

= -(n- m- 1)(m- 1) 100 sinhm-a r(Mr f)(p) dr.

Applying this repeatedly to (29) we obtain Theorem 1.5.

B. The Spheres and the Elliptic Spaces

Now let X be the unit sphere sn(o) c JR1l+l and S the set of k-dimensional totally geodesic submanifolds of X. Each ~ E S is a k-sphere. We shall now invert the Radon transform

f(~) = i f(x) dm(x)' IE e(X)

where dm is the measure on ~ given by the Riemannian structure induced by that of X. In contrast to the hyperbolic space, each geodesic X through a point x also passes through the antipodal point Ax. As a result, 1 = (! o A) and our inversion formula will reflect this fact. Although we state our result for the sphere, it is really a result for the elliptic space, that is the sphere with antipodal points identified. The functions on this space are naturally identified with symmetric functions on the sphere.

Again let

denote the average of a continuous function on S over the set of ~ passing through x.

Theorem 1.7. Let k be an integer, 1 $ k < n =dim X.

{i) The mapping I ~ 1 (! E e(X)) has kernel consisting of the skew function {the functions f satisfying f + f o A= 0}.

{ii} Assume k even and let P~r denote the polynomial

H:(z) = [z-(k-1)(n-k)][z- (k-3)(n-k+2)] ... [z-l(n-2)]


of degree k/2. The k-dimensional Radon transform on X is then inverted by the formula

where c is the constant in (27).

Proof. We first prove (ii) in a similar way as in the noncompact case. The Riemannian structure in (3) is now replaced by

ds2 = dr2 + sin2 r da2 ;

the Laplace-Beltrami operator is now given by

82 8 (3I) L = 8r2 + (n- I) cot r 8r +(sin r)-2 Ls

instead of {23) and

(j)V(x) = flk fotr (Mr f)(x) sink-l rdr.

For a fixed x we put F{r) = (Mr f)(x). The analog of Lemma 1.6 now reads as follows.

Lemma 1.8. Let m be an integer, 0 < m < n = dim X. Then

ltr sinmrLrFdr =

(n- m -I) [m 1tr sinm rF(r) dr- (m -I) 1tr sinm-2 rF(r) dr]

If m = I, the term (m - I) J0tr sinm-2 rF(r) dr should be replaced by F(o) + F(1r).

Since (30) is still valid the lemma implies

[La: - m(n- m- I)] 11f sinm r(Mr f)(x) dr

= -(n- m- I){m- 1) 1tr sinm-2 r(Mr f)(x) dr

and the desired inversion formula follows by iteration since

F{O) + F(1r) = f(x) + f(Ax).

In the case when k is even, Part (i) follows from (ii). Next suppose k = n- 1, n even. For each e there are exactly two points X and Ax at maximum distance, namely ~' from e and we write

[(x) =[(Ax) =[(e).


We have then

(32)

Next we recall some well-known facts about spherical harmonics. We have

(33)

where the space 1is consist of the restrictions to X of the homogeneous harmonic polynomials on JRn+l of degrees.

(a) Lh8 = -s(s + n- 1)h8 (hs E 1is) for each s ;:::: 0. This is immediate from the decomposition

{)2 no 1 Ln+I=-+--+-L or2 r or r 2

of the Laplacian Ln+l of JRn+l ( cf. (23)). Thus the spaces 1is are precisely the eigenspaces of L.

(b) Each 1is contains a function(~ 0) which is invariant under the group K ofrotations around the vertical axis (the Xn+1-axis in JRn+l ) . This function <p8 is nonzero at the North Pole o and is uniquely determined by the condition <p8 (o) = 1. This is easily seen since by (31) <p8 satisfies the ordinary differential equation

~~ ~s ) dr2 + (n- 1) cot r dr = -s(s + n- 1 I{Js, <p~(o) = 0.

It follows that 1is is irreducible under the orthogonal group O(n + 1).

(c) Since the mean value operator M1r 12 commutes with the action of O(n + 1) it acts as a scalar c8 on the irreducible space 1i8 • Since we have

M1r/2 1.p8 = C8 i.p8 , <p8 (o) = 1,

we obtain

(34) Cs = I{Js(~) ·

Lemma 1.9. The scalar <p8 (7r/2) is zero if and only if s is odd.

Proof. Let H8 be the K-invariant homogeneous harmonic polynomial whose restriction to X equals <p8 • Then Hs is a polynomial in xi+···+ x;

96 Chapter ITI. The Radon Transform on Two-point Homogeneous Spaces

and Xn+l so if the degree 8 is odd, Xn+l occurs in each term whence t.p8 (7r /2) = H8 (I, 0, ... 0, 0) = 0. H 8 is even, say 8 = 2d, we write

Using Ln+l = Ln + 82 f8x~+l and formula (3I) in Ch. I the equation Ln+lHs = 0 gives the recursion formula

ai(2d- 2i)(2d- 2i + n- 2) + ai+l (2i + 2)(2i +I) = 0

(0 ~ i < d). Hence H8 (1, 0 ... 0), which equals ao, is "# 0; Q.e.d.

Now each f E &(X) has a uniformly convergent expansion

and by (32)

If 1 = 0 then by Lemma 1.9, h8 = 0 for 8 even so f is skew. Conversely 1 = 0 iff is skew so Theorem 1.7 is proved for the case k = n- I, n even.

H k is odd, 0 < k < n- 1, the proof just carried out shows that J(~)=O for all ~ E E implies that f has integral 0 over every (k + I)-dimensional sphere with radius 1 and center o. Since k + 1 is even and < n we conclude by (ii) that f + f o A = 0 so the theorem is proved.

As a supplement to Theorems 1.5 and 1. 7 we shall now prove an inversion formula for the Radon transform for general k (odd or even).

Let X be either the hyperbolic space Hn or the sphere sn and E the space of totally geodesic submanifolds of X of dimension k (1 ::; k::; n- 1). We then generalize the transforms f -+ 1, t.p -+ (/; as follows. Let p 2:: 0. We put

(35) h(~) = { f(x) dm(x), Jd(z,~}=p

where dm is the Riemannian measure on the set in question and df.-t is the average over the set of eat distance p from X. Let ep be a fixed element of E at distance p from 0 and Hp the subgroup of G leaving ~P stable. It is then easy to see that in the language of Ch. II, §I

(36) X = gK' e = 'YHp are incident {::} d(x, e) = p.


This means that the transforms {35) are the Radon transform and its dual for the double fibration

GJK

For X = S2 the set {X : d{ X' e) = p} is a circle on S2 of length 211" cos p. For X= H 2 , the non-Euclidean disk, e a diameter, the set {x: d{x,~) = p} is a circular arc with the same endpoints as e. Of course fo = j, 'Po= ip.

We shall now invert the transform f -+ f by invoking the more general transform cp -+ 'Pv· Consider x E X,~ E S with d(x, ~) = p. Select g E G such that g · o = x. Then d(o,g-1~) = p so {kg-1 • ~: k E K} is the set of "1 E Sat distance p from o and {gkg-1 • ~: k E K} is the set of fJ E Sat distance p from x. Hence

so

{37)

(f)~ (g. o) = L f{gkg- 1 • e) dk = L dk l f(gkg- 1 . y) dm(y)

= l (L f(gkg- 1 • y) dk) dm(y)

FIGURE III.2.

Let Xo E e be a point at minimum distance (i.e., p) from x and let (Fig. III.2)

{38) r=d(xo,y), q=d(x,y), yE~.

Since e c X is totally geodesic, d( Xo' y) is also the distance between X 0 and y in e. In (37) the integrand (Md(x,y ) f)(x) is constant in y on each sphere in ' with center X 0 •

Theorem 1.10. The k-dimensional totally geodesic Radon transform f -+ f on the hyperbolic space Hn is inverted by

f(x) = c [ ( d(~2)) k lou (j)(ro v(x)(u2 - v2)~-1 dv] u=1'

where c- 1 = (k -l)!f2Hl/2k+l, 1m v = cosh-1(v-1).


Proof. Applying geodesic polar coordinates in ~ with center x0 we obtain from {37)-{38),

(39) (f)~ (x) = Ok 100 (Mq !)(x) sinhk-1 r dr.

Using the cosine relation on the right-angled triangle (xx0 y) we have by {38) and d(xo, x) = p,

(40) cosh q = cosh p cosh r .

With x fixed we define F and F by

{41) F(coshq) = (Mqf)(x), F(coshp) = (J)~(x).

Then by (39),

(42) F(coshp) = nk 100 F(coshpcoshr) sinhk-1 rdr.

Putting here t = coshp, s = cosh r this becomes

F(t) = nk 100 F(ts)(s2 -1)~-1 ds'

which by substituting u = (ts)-1 ' v = r 1 becomes

v-1 F(v- 1 ) = nk 11J F(u-1 )u-k(v2 - u2)~-1 du.

This is of the form (19), Ch. I, §2 and is inverted by

(43) F(u-1)u-k = cu(-d-)k r· (u2 - v2)~-1 F(v-1 ) dv d(u2 ) } 0 '

where c-1 = (k - 1)0k+tf2k+l. Defining 1m v by cosh(lm v) = v-1

and noting that f(x) = F(coshO) the theorem follows by putting u = 1 in (43).

For the sphere X = sn we can proceed in a similar fashion. We assume f symmetric (f(s) = f( -s)) because f = 0 for f odd. Now formula (37) takes the form

(44) (f)~ (x) = 20k 1i (Mq f)(x) sink-1 r dr,

(the factor 2 and the limit 7r/2 coming from the symmetry assumption). This time we use spherical trigonometry on the triangle (xx0 y) to derive

cosq = cospcosr.


We fix x and put

(45) F(cosq) = (Mq f)(x), F(cosp) =(f)~ (x).

and

v=cosp, u=vcosr.

Then ( 44) becomes

(46) vk-1F(v) = 2!lk 1v F(u)(v2 - u2)~-l du,

which is inverted by

F(u) = ~u(d(~2)) k 1u (u2 - v2)~-lvk F(v) dv,

c being as before. Since F(l) = f(x) this proves the following analog of Theorem 1.10.

Theorem 1.11. The k-dimensional totally geodesic Radon transform f -+ i on sn is for f symmetric inverted by

c ~ v k 2 2 .&__1 [( d ) k 1u ] f(x) = 2 d(u2) o (f)cos-l(v)(x)v (u - v ) 2 dv u=l

where

Geometric interpretation

In Theorems 1.1Q-l.ll, (j)~(x) is the average of the integrals off over the k-dimensional totally geodesic submanifolds of X which have distance p from x.

We shall now look a bit closer at the geometrically interesting case k = 1. Here the transform f -+ f is called the X-ray transform.

We first recall a few facts about the spherical transform on the constant curvature space X = G I K, that is the hyperbolic space Hn = Q::. or the sphere sn = Q +. A spherical junction <p on G I K is by definition a K-invariant function which is an eigenfunction of the Laplacian L on X satisfying <p( o) = 1. Then the eigenspace of L containing <p consists of the functions f on X satisfying the functional equation

(47) i f(gk · x) dk = f(g · o)<p(x)


([GGA), p. 64). In particular, the spherical functions are characterized by

{48) [ cp(gk · x) dk = cp(g · o)cp(x) cp ~ 0.

Consider now the case H2 • Then the spherical functions are the solutions cpA(r) of the differential equation

(49) ~cpA dcpA 2 1 dr2 +cothr dr =-(A +4)cpA, cpA(o)=l.

Here A E C and 'P-A = 'PA· The function cpA has the integral representation

(50) cpA(r) =- (chr- shrcosO)-t.X+2 dO. 111r . 1

7r 0

In fact, already the integrand is easily seen to be an eigenfunction of the operator Lin (23) (for n = 2) with eigenvalue -(A2 + 1/4).

H f is a radial function on X its spherical transform J is defined by

(51)

for all A E C for which this integral exists. The continuous radial functions on X form a commutative algebra C~(X) under convolution

(52) (!1 x h)(g · o) = L !l(gh-1 • o)h(h · o) dh

and as a consequence of (48) we have

(53)

In fact,

(!1 x h)~(A) = fa !l(h · o) (fa h(g · o)cp-.x(hg · o) dg) dh

= Lh(h·o) (Lh(g·o)) ([ 'P-A(hkg·o)dkdg) dh

= h(A)];(A).

We know already from Corollary 1.3 that the Radon transform on Hn is injective and is inverted in Theorem 1.5 and Theorem 1.10. For the case n = 2, k = 1 we shall now obtain another inversion formula based on (53).

The spherical function cpA(r) in (50) is the classical Legendre function Pv (cosh r) with v = iA - t for which we shall need the following result ([Prudnikov, Brychkov and Marichev), Vol. Ill, 2.17.8(2)).

§1 Spaces of Constant Curvature. Inversion and Support Theorems Hil

Lemma 1.12.

(54)

for

(55) Re(p - v) > 0 , Re(p + v) > -1.

We shall require this result for p = 0, 1 and ,\real. In both cases, conditions (55) are satisfied.

Let r and u denote the functions

(56) r(x) =sinh d(o,x)- 1 , u(x) = coth(d(o,x)) -1, x EX.

Lemma 1.13. For f E V(X) we have

(57)

Proof. In fact, the right hand side is

r sinh d(x, y)-1 f(y) dy = rX) dr(sinh r)-1 r f(y) dw(y) h h kw

so the lemma follows from {28). Similarly we have

(58) Sf=fxu,

where S is the operator

{59) (Sf)(x) = i (coth(d(x, y)) -l)f(y) dy.

Theorem 1.14. The operator f -t f is inverted by

(60)

Proof. The operators ~ , v, S and L are all G-invariant so it suffices to verify {60) at o. Let Jb(x) = JK f(k · x) dk. Then

{f X r)b = Jb X T, {f X u)b = Jb Xu, (Lf)(o) = (£/~)(o).

Thus by (57)-(58)

LS((f}v)(o) = L(S(([)V))b(o) = L{f x r x u)b(o) = LS(((Jbf)v) )(o).


Now, if (60) is proved for a radial function this equals cf'rl(o) = cf(o). Thus (60) would hold in general. Consequently, it suffices to prove

(61) L(f x T xu) = -471"2 f, f radial in V(X).

Since f, T and u are all integrable on X,

(62) (!X T X 17)~(.\) = 1(.\)r(.\)a(.\).

Since cothr -1 = e-r /sinh r, and since dx =sinh rdrdO, r(.\) and a(.\) are given by the left hand side of (54) for p = 0 and p = 1, respectively. Thus

r(.\)

r(1 _ u)r(iA + 1) a(.\) _ 7r 4 2 2 4

- rei;+ ~)r(~- i;).

Using the identity r(x + 1) = xr(x) on the denominator of a(.\) we see that

(63)

Now since

L(f X T Xu) = (Lf X T Xu), f E 1J'rl(X),

and since by {49), (Lf)~(.\) = -(.\2 + t)f(.\), we deduce from (62)-(63) that (60) holds with the constant -47r2 •

It is easy to write down an analog of (60) for S2 • Let o denote the North Pole and put

r(x) = sind(o,x)-1 x E S2 .

Then in analogy with (57) we have

(64)

where x denotes the convolution on 82 induced by the convolution on G. The spherical functions on G / K are the functions

'Pn(x) = Pn(cos d(o, x)) n ~ 0,

where Pn is the Legendre polynomial

1 {27r Pn(cosO) = 27r Jo (cosO+isinOcosu)ndu.


Since Pn(cos(7r- 8)) = (-1)nPn(cos8), the expansion ofT into spherical functions

00

r(x) "'(4n + 1) L7(2n)P2n(cos d(o, x)) n=O

only involves even indices. The factor ( 4n + 1) is the dimension of the space of spherical harmonics containing <{)2n· Here the Fourier coefficient r(2n) is given by

r(2n) = 41 r r(x)c.p2n(x) dx, 1r ls2

which, since dx = sin 8 d8 dc.p, equals

(65) 1 1,. 1r (2n) 2

411" 211" 0

P2n( COS 8) d8 = 24n-l n ,

by loc. cit., Vol. 2, 2.17.6 (11). We now define the functional a on S2 by the formula

(66)

where

(67)

00

a(x) = (4n + 1) :La2nP2n(cos d(o,x)), 0

To see that (66) is well-defined note that

( 2nn) = 2n1. 3·· · (2n -1)/n! 2: 2n1. 2 ·4· ·· (2n- 2)/n!

> 22n-1/n

so a2n is bounded in n. Thus a is a distribution on S2 . Let S be the operator

(68) Sf=fxa.

Theorem 1.15. The operator f-+ f is inverted by

(69)

104 Chapter III. The Radon 'Iransform on Two-point Homogeneous Spaces

Proof. Just as is the case with Theorem 1.14 it suffices to prove this for f Kinvariant and there it is a matter of checking that the spherical transforms on both sides agree. For this we use (64) and the relation

LIP2n = -2n(2n + 1)!1'2n.

Since

(r X u)~(2n) = r(2n)a2n.

the identity (69) follows.

A drawback of (69) is of course that (66) is not given in closed form. We shall now invert f -+ j in a different fashion on S2 • Consider the spherical coordinates of a point ( x17 x2, x3) E S2 •

(70) XI = COS lp Sin(}, X2 = Sin lp sin(}, X3 = COS(}

and let kcp = K denote the rotation by the angle IP around the x3-axis. Then f has a Fourier expansion

f(x) = L fn(x), nEZ

1 r7r . fn(x) = 27r Jo f(kcp · x)e-mcp dip.

Then

for each great circle 'Y. In particular, f n is determined by its restriction 9 = fnlx1=0• i.e.,

g(cosO) = fn(O,sinO,cosO).

Since fn is even, (70) implies g(cos(1r- 8)) = (-1)ng(cos0), so

g(-u) = (-l)ng(u).

Let r be the set of great circles whose normal lies in the plane XI = 0. H 'Y E r let X-y be the intersection of 'Y with the half-plane XI = 0, X2 > 0 and let a be the angle from o to X-y, (Fig. III.3). Since fn is symmetric,

(72)

where x 8 is the point on 'Y at distance 8 from x-y (with XI (xs) ~ 0). Let !p

and () be the coordinates (70) of X 8 • Considering the right angled triangle X8 0X-y we have

cos () = cos 8 cos a


I I I I I

I e I I I I I I I ,.-."" ............

......... ;

I I I I I I I

•

FIGURE III.3.

and since the angle at o equals 1rj2- t.p, (71) implies

g(cosa) = fn(x7 ) = ein(1rj2-tp) fn(xa).

Writing

(73) g(cosa) = fnb)

equation (72) thus becomes

g(cosa) =2(-i)n 11r ein'Pg(cosB)ds.

Put v = cos a , u - v cos s, so

du = v(- sins) ds = -(v2 - u2 ) 112 ds.

Then

(74) g(v) = 2( -it/_: einip(u,v) g(u)(v2 - u2)-t du,


where the dependence of IP on u and v is indicated (for v :f. 0). Now -u = v cos( 1r - s) so by the geometry, IP( -u, v) = -~P( u, v). Thus

(74) splits into two Abel-type Volterra equations

(75) g(v) = 4( -l)nl2 1v cos(nip(u, v))g(u)(v2 - u 2)-! du, n even

(76) g(v) = 4(-l)(n-l)/2 1v sin(nip(u,v))g(u)(v2 - u2)-! du, n odd.

For n = 0 we derive the following result from ( 43) and (75).

Proposition 1.16. Let f E C2 (S2) be symmetric and K -invariant and fits X-ray transform. Then the restriction g(cosd(o,x)) = f(x) and the function g( cos d( o, "f)) = f( "f) are related by

(77) g(v) = 41v g(u)(v2 - u2)-! du

and its inversion

(78) 21rg(u) = :U 1u g(v)(u2 - v2)-t vdv.

We shall now discuss the analog for sn of the support theorem (Theorem 1.2) relative to the X-ray transform f-+ f. Theorem 1.17. Let C be a closed spherical cap on sn, C' the cap on sn symmetric to C with respect to the origin 0 E JRn+l. Let f E C(Sn) be symmetric and assume

(79) fc'Y) = 0

for every geodesic 'Y which does not enter the "arctic zones" C and C'. (See Fig. II.3.)

(i) If n ~ 3 then f = 0 outside C U C'.

{ii} If n = 2 the same conclusion holds if all derivatives of f vanish on the equator.

Proof. (i) Given a point X E sn outside cue' we can find a 3-dimensional subspace ~ of JRn+l which contains x but does not intersect C U C'. Then ~ n sn is a 2-sphere and f has integral 0 over each great circle on it. By Theorem 1.7, f:::: 0 on~ n sn so f(x) = 0.

(ii) If f is K-invariant our statement follows quickly from Proposition 1.16. In fact, if C has spherical radius /3, (79) implies g(v) = 0 for 0 < v < cos/3 so by (78) g(u) = 0 for 0 < u < cos/3 so f = 0 outside C U C'.


FIGURE III.4.

Generalizing this method to fn in (71) by use of (75)-(76) runs into difficulties because of the complexity of the kernel ein<p(u,v) in (74) near v = 0. However, if f is assumed = 0 in a belt around the equator the theory of the Abel-type Volterra equations used on (75)-(76) does give the conclusion of (ii). The reduction to the K-invariant case which worked very well in the proof of Theorem 1.2 does not apply in the present compact case.

A better method, due to Kurusa, is to consider only the lower hemisphere s:_ of the unit sphere and its tangent plane 1r at the South Pole S. The central projection J.l- from the origin is a bijection of s:_ onto 1r which intertwines the two Radon transforms as follows: If 1 is a (half) great circle on s:_ and i the line J..l-b) in 1r we have (Fig. III.4)

The proof follows by elementary geometry: Let on Fig. III.4, x = J.l-(s), r.p and e the lengths of the arcs S M, Ms. The plane d S o11 is perpendicular to i and intersects the semi-great circle 1 in M. If q = jSo''j,p = jo11xl we have for f E C(S2 ) symmetric,

~ l 1 dO J(/)=2 f(s)d8=2 (fof.1-- 1)(x)ddp. "Y l p

108 Chapter III. The Radon 'fransform on Two-point Homogeneous Spaces

Now

tan<p = q, p

tan8 = (1 + q2)1/2 '

so

Thus

and since <p = d(S, "f) this proves (80). Considering the triangle o' xS we obtain

(81) lxl = tand(S, s).

Thus the vanishing of all derivatives of f on the equator implies rapid decrease off o f.t-1 at oo.

Now if <p > f3 we have by assumption, f("f) = 0 so by (80) and Theorem 2.6 in Chapter I,

(! o f.t-1)(x) = 0 for lxl > tan/3,

whence by (81),

f(s) = 0 for d(S, s) > f3.

Remark 1.18. Because of the example in Remark 2.9 in Chapter I the vanishing condition in (ii) cannot be dropped.

There is a generalization of (80) to d-dimensional totally geodesic submanifolds ofSn as well as ofHn (Kurusa (1992], [1994], Berenstein-Tarabusi [1993]). This makes it possible to transfer the range characterizations of the d-plane Radon transform in IRn (Chapter I, §6) to the d-dimensional totally geodesic Radon transform in Hn. In addition to the above references see also Berenstein-Tarabusi-Kurusa [1997], Gindikin [1995] and Ishikawa [1997].

C. The Spherical Slice Transform

We shall now briefly consider a variation on the Funk transform and consider integrations over circles on S2 passing through the North Pole. This Radon transform is given by j -+ f where j is a function on S2 ,

(82) f("f) = i f(s) dm(s),


N

FIGURE III.5.

'Y being a circle on 82 passing through N and dm the arc-element on 'Y. It is easy to study this transform by relating it to the X-ray transform

on JR2 by means of stereographic projection from N. We consider a two-sphere 82 of diameter 1, lying on top of its tangent

plane JR2 at the South Pole. Let 11 : 82 - N -t JR2 be the stereographic projection. The image v('Y) is a line i c JR2 • (See Fig. ill.5.) The plane through the diameter N S perpendicular to i intersects 'Y in so and i in xo. Then N so is a diameter in 'Y, and in the right angle triangle N S x0 , the line Ss0 is perpendicular to Nx0 • Thus, d denoting the Euclidean distance in JR3 , and q = d(S, xo), we have

Let a denote the circular arc on 'Y for which v(a) is the segment (xo, x) on i. If e is the angle between the lines N Xo, N X then


Thus, dm(x) being the arc-element on l,

dm(x) du

Hence we have

(85)

a formula quite similar to {80). Iff lies on C1{S2 ) and vanishes at N then f o v-1 = O(x-1 ) at oo. Also

off E E(S2) and all its derivatives vanish at N then fov- 1 E S(R.2 ). As in the case of Theorem 1.17 (ii) we can thus conclude the following corollaries of Theorem 3.1, Chapter I and Theorem 2.6, Chapter I.

Corollary 1.19. The transform f -t 1 is one-to-one on the space CJ(S2 )

of C1 -functions vanishing at N.

In fact, (f o v-1 )(x)/(1 + lxl2 ) = O(lxl-3) so Theorem 3.1, Chapter I applies.

Corollary 1.20. Let B be a spherical cap on 8 2 centered at N. Let f E C00 (S2) have all its derivatives vanish at N. If

1b) = 0 for all 1 through N , 1 C B

then f = 0 on B.

In fact (f o v-1)(x) = O(lxl-k) for each k ~ 0. The assumption on 1 implies that (f o v-1 )(x)(1 + lxl2)-1 has line integral 0 for all lines outside v(B) so by Theorem 2.6, Ch. I, f o v-1 = 0 outside v(B).

Remark 1.21. In Cor. 1.20 the condition of the vanishing of all derivatives at N cannot be dropped. This is clear from Remark 2.9 in Chapter I where the rapid decrease at oo was essential for the conclusion of Theorem 2.6.

If according to Remark 3.3, Ch. I g E E(R.2 ) is chosen such that g(x) = O(jxj-2 ) and all its line integrals are 0, the function f on 82 - N defined by

is bounded and by (85), f{"Y) = 0 for all "Y· This suggests, but does not prove, that the vanishing condition at N in Cor. 1.19 cannot be dropped.

§2 Compact Two-point Homogeneous Spaces. Applications 111

§2 Compact Two-point Homogeneous Spaces. Applications

We shall now extend the inversion formula in Theorem 1. 7 to compact two-point homogeneous spaces X of dimension n > 1. By virtue of Wang's classification [1952] these are also the compact symmetric spaces of rank one (see Matsumoto [1971] and Szabo [1991] for more direct proofs), so their geometry can be described very explicitly. Here we shall use some geometric and group theoretic properties of these spaces ((i)-(vii) below) and refer to Helgason ([1959], p. 278, (1965a], §5-6 or [DS], Ch. VII, §10) for their proofs.

Let U denote the group I(X) of isometries X. Fix an origin o EX and let K denote the isotropy subgroup U0 • Let t and u be the Lie algebras of K and U, respectively. Then u is semisimple. Let p be the orthogonal complement of t and u with respect to the Killing form B of u. Changing the distance function on X by a constant factor we may assume that the differential of the mapping u -+ u · o of U onto X gives an isometry of p (with the metric of -B) onto the tangent space X 0 • This is the canonical metric X which we shall use.

Let L denote the diameter of X, that is the maximal distance between any two points. If x E X let Ax denote the set of points in X of distance L from x. By the two-point homogeneity the isotropy subgroup Ux acts transitively on Ax; thus Ax C X is a submanifold, the antipodal manifold associated to x.

(i) Each Ax is a totally geodesic submanifold of X; with the Riemannian structure induced by that of X it is another two-point homogeneous space.

(ii) Let 2 denote the set of all antipodal manifolds in X; since U acts transitively on 2, the set 2 has a natural manifold structure. Then the mapping j : x -+ Ax is a one-to-one diffeomorphism; also x E Ay if and only if y E Ax.

(iii) Each geodesic in X has period 2L. If x EX the mapping Expx : Xx-+ X gives a diffeomorphism of the ball BL(O) onto the open set X- Ax.

Fix a vector HE p of length L (i.e., L 2 = -B(H, H)). For Z E plet Tz denote the linear transformation Y-+ [Z, [Z, Y]J of p, [,] denoting the Lie bracket in u. For simplicity, we now write Exp instead of Exp0 • A point Y E }l is said to be conjugate to o if the differential dExp is singular at Y.

The line a= lRH is a maximal abelian subspace of p. The eigenvalues of TH are 0, a(H) 2 and possibly (a(H)/2)2 where ±a (and possibly ±a/2) are the roots of u with respect to a. Let

(86) }l = a+ Pa + l'a/2


be the corresponding decomposition of p into eigenspaces; the dimensions q = dim(Pa), p = dim(Pa;2 ) are called the multiplicities of a and a/2, respectively.

(iv) Suppose His conjugate too. Then Exp(n+Pa), with the Riemannian structure induced by that of X, is a sphere, totally geodesic in X, having o and ExpH as antipodal points and having curvature 1r2 L 2. Moreover

(v) If H is not conjugate too then Pa;2 = 0 and

AExpH = Exp Pa ·

(vi) The differential at Y of Exp is given by

oo Tk dExpy = dr(exp Y) o ~ {2k: I)! ,

where for u E U, r(u) is the isometry x-+ u · x.

(vii) In analogy with (23} the Laplace-Beltrami operator L on X has the expression

fJ2 1 a L = 8r2 + A(r) A'(r) 8r + Ls.,.'

where Ls.,. is the Laplace-Beltrami operator on Sr(o) and A(r) its area.

(viii) The spherical mean-value operator Mr commutes with the LaplaceBeltrami operator.

Lemma 2.1. The surface area A(r) (0 < r < L) is given by

A(r) = f2nA -P(2.\)-q sinP(.\r) sinq(2.\r)

where p and q are the multiplicities above and A= la(H)I/2L.

Proof. Because of {iii) and (vi) the surface area of Br(o) is given by

1 oo Tk A(r) = det(L (2k Y !) 1)dwr(Y),

IYI=r o + · where dwr is the surface on the sphere IYI = r in p. Because of the two-point homogeneity the integrand depends on r only so it suffices to evaluate it for Y = Hr = fH. Since the nonzero eigenvalues of Tn.,. are a(Hr)2 with multiplicity q and (a(Hr)/2)2 with multiplicity p, a trivial computation gives the lemma.


We consider now Problems A, B and C from Chapter II, §2 for the homogeneous spaces X and 3, which are acted on transitively by the same group U. Fix an element eo E 3 passing through the origin o E X. If ~o = Ao, then an element u E U leaves eo invariant if and only if it lies in the isotropy subgroup K' = Uo; we have the identifications

X = u I K' 3 = u I K'

and x E X and e E 3 are incident if and only if x E e. On 3 we now choose a Riemannian structure such that the diffeomor

phism j : x -4 Ax from (ii) is an isometry. Let L and A denote the Laplacians on X and 3, respectively. With x and [defined as inCh. II, §1, we have

f = ~, x = {j(y) : y E j(x)};

the first relation amounts to the incidence description above and the second is a consequence of the property x E Ay {:} y E Ax listed under (ii).

The sets x and [ will be given the measures df.t and dm, respectively, induced by the Riemannian structures of 3 and X. The Radon transform and its dual are then given by

f{~) = h f(x) dm(x), <P(x) = h cp(e) df.t(~).

However

<P(x) = ~cp{e)dJ.t(e) = r cp(j(y))dJ.t(j(y)) = r (cpoj)(y)dm(y) lx JYEi(x) Ji(x)

so

(87) <P = ( cp 0 jf 0 j .

Because of this correspondence between the transforms f -4 j, cp -4 i:p it suffices to consider the first one. Let D(X) denote the algebra of differential operators on X, invariant under U. It can be shown that D(X) is generated by L. Similarly D(3) is generated by A.

Theorem 2.2. (i) The mapping f -4 f is a linear one-to-one mapping of £(X) onto £(3) and

(Lff = Aj.

(ii) Except for the case when X is an even-dimensional elliptic space

f = P(L)((j)v), f E £(X),


where P is a polynomial, independent off, explicitly given below, (90}(93}. In all cases

degree P = ! dimension of the antipodal manifold.

Proof. (Indication.) We first prove (ii). Let dk be the Haar measure on K such that J dk = 1 and let Ox denote the total measure of an antipodal manifold in X. Then J.l.(OJ = m(Ao) =Ox and if u E U,

cp(u. o) = nx L <p(uk. ~o) dk.

Hence

(j)v(u·o) =Ox 1 (l f(uk·y)dm(y)) dk=Ox l (Mr!)(u·o)dm(y), K ~o ~o

where r is the distance d( o, y) in the space X between o and y. If d( o, y) < L there is a unique geodesic in X of length d( o, y) joining o to y and since ~0 is totally geodesic, d( o, y) is also the distance in ~0 between o and y. Thus using geodesic polar coordinates in ~o in the last integral we obtain

(88)

where A1(r) is the area of a sphere of radius r in ~O· By Lemma 2.1 we have

(89)

where C1 and ..\1 are constants and Pl. q1 are the multiplicities for the antipodal manifold. In order to prove (ii) on the basis of (88) we need the following complete list of the compact symmetric spaces of rank one and their corresponding antipodal manifolds:

X Ao

Spheres sn(n = 1, 2, ... ) point

Real projective spaces pn(JR)(n = 2, 3, ... ) pn-l(JR)

Complex projective spaces pn(C)(n = 4, 6, ... ) pn-2(C)

Quaternian projective spaces pn(H)(n = 8, 12, ... ) pn-4(H)

Cayley plane p16(Cay) ss

Here the superscripts denote the real dimension. For the lowest dimensions, note that


For the case sn, (ii) is trivial and the case X = pn (JR.) was already done in Theorem 1. 7. The remaining cases are done by classification starting with (88). The mean value operator Mr still commutes with the Laplacian L

MrL=LMr

and this implies

where Lr is the radial part of L. Because of (vii) above and Lemma 2.1 it is given by

a2 a Lr = ar2 + .X{pcot(.Xr) + 2qcot(2.Xr)} ar.

For each of the two-point homogeneous spaces we prove (by extensive computations) the analog of Lemma 1.8. Then by the pattern of the proof of Theorem 1.5, part (ii) of Theorem 2.2 can be proved. The full details are carried out in Helgason ([1965a] or [GGA], Ch. I, §4).

The polynomial P is explicitly given in the list below. Note that for pn (JR.) the metric is normalized by means of the Killing form so it differs from that of Theorem 1.7 by a nontrivial constant.

The polynomial P is now given as follows: For X = pn (JR.), n odd

(90) P(L) = c(L- <n;~> 1 ) (L- (n;,;l3 ) ... (£- 1(~~2)) 1

c = H-41l'2n)2(n-1).

For X= pn(C), n = 4,6,8, ...

(91) P(L) (L ~) (L ~) (L ~) = C - 2(1l-F2J - 2(n+2) · · · - 2(n+2)

c = (-811'2 (n+2))1-¥. For X= pn(H), n = 8, 12, ...

(92) P(L) (L (n-2)4) (L (n-4)6) (L ~) = C - 2'{n+8) - 2'{n+8) · · • - 2 ( n+8)

c = H-411'2(n + 8W-nf2.

For X= P 16 (Cay)

That f -t J is injective follows from (ii) except for the case X = pn (JR.), n even. But in this exceptional case the injectivity follows from Theorem 1.7.


For the surjectivity we use once more the fact that the mean-value operator Mr commutes with the Laplacian (property (viii)). We have

(94) 1U(x)) = c(ML f)(x),

where c is a constant. Thus by (87)

so

(95)

Thus if X is not an even-dimensional projective space f is a constant multiple of MLP(L)MLJ which by (94) shows f-+ 1 surjective. For the remaining case pn(JR), n even, we use the expansion off E t'(Pn(JR)) in spherical harmonics

f = L akmSkm (k even). k,m

Here k E z+, and Skm(l ~ m ~ d(k)) is an orthonormal basis of the space of spherical harmonics of degree k. Here the coefficients akm are rapidly decreasing in k. On the other hand, by (32) and (34),

(96) 1 = OnM~ f =On Lakmft'k (~) Skm (k even). k,m

The spherical function fPk is given by

ft'k(s) = 0~:1 11T (cosO+ i sinO coscp)k sinn-2 cpdcp

so fP2k(~) "'k-~. Thus every series l:k,m bk,mS2k,m with b2k,m rapidly decreasing in k can be put in the form {96). This verifies the surjectivity of the map f -+ J.

It remains to prove (Lff= Aj. For this we use (87), (vii), (41) and (94). By the definition of A we have

(Acp)(j(x)) = L(cp o j)(x), x E X,cp E t'(X).

Thus

(A1)(j(x)) = (L(1 o j))(x) = cL(ML f)(x) = cML(Lf)(x) = (Lj)(j(x)).

This finishes our indication of the proof of Theorem 2.2.


Corollary 2.3. Let X be a compact two-point homogeneous space and suppose f satisfies

i f(x) ds(x) = 0

for each (closed) geodesic 1 in X, ds being the element of arc-length. Then

(i) If X is a sphere, f is skew.

(ii) If X is not a sphere, f = 0.

Taking a convolution with f we may assume f smooth. Part (i) is already contained in Theorem 1.7. For Part (ii) we use the classification; for X = P 16 (Cay) the antipodal manifolds are totally geodesic spheres so using Part (i) we conclude that j = 0 so by Theorem 2.2, f = 0. For the remaining cases pn(C) (n = 4, 6, ... ) and pn(H), (n = 8, 12, ... ) (ii) follows similarly by induction as the initial antipodal manifolds, P 2 (C) and P 4 (H), are totally geodesic spheres.

Corollary 2.4. Let B be a bounded open set in ]Rn+l, symmetric and starshaped with respect to 0, bounded by a hypersurface. Assume for a fixed k (1 ~ k < n)

(97) Area (B n P) = constant

for all (k +!)-planes P through 0. Then B is an open ball.

In fact, we know from Theorem 1. 7 that if f is a symmetric function on X = sn with [(sn n P) constant (for all P) then f is a constant. We apply this to the function

f(8) = p(8)k+l 8 E sn

if p( e) is the distance from the origin to each of the two points of intersection of the boundary of B with the line through 0 and 8; f is well defined since B is symmetric. If e = (81 , ... , fh) runs through the k-sphere sn n P then the point

x=8r (O~r<p(O))

runs through the set B n P and

l 1p(e) Area (B n P) = dw(O) rk dr.

S"nP 0

It follows that Area (B n P) is a constant multiple of [(sn n P) so (97) implies that f is constant. This proves the corollary.


§3 Noncompact Two-point Homogeneous Spaces

Theorem 2.2 has an analog for noncompact two-point homogeneous spaces which we shall now describe. By Tits' classification [1955], p. 183, of homogeneous manifolds L I H for which L acts transitively on the tangents to L I H it is known, in principle, what the noncom pact two-point homogeneous spaces are. As in the compact case they turn out to be symmetric. A direct proof of this fact was given by Nagano [1959] and Helgason [1959]. The theory of symmetric spaces then implies that the noncompact twopoint homogeneous spaces are the Euclidean spaces and the noncompact spaces X = G I K where G is a connected semisimple Lie group with finite center and real rank one and K a maximal compact subgroup.

Let g = t + p be the direct decomposition of the Lie algebra of G into the Lie algebra t of K and its orthogonal complement p (with respect to the Killing form of g). Fix a !-dimensional subspace a C p and let

(98) p = a+ Pa + l'a/2

be the decomposition ofp into eigenspaces ofTH (in analogy with (86)). Let ~0 denote the totally geodesic submanifold Exp(Pa;2); in the case Pa;2 = 0 we put eo = Exp(Pa)· By the classification and duality for symmetric spaces we have the following complete list of the spaces G I K. In the list the superscript denotes the real dimension; for the lowest dimensions note that

X & Real hyperbolic spaces Hn(JR.)(n = 2, 3, ... ), Hn-1 (JR.)

Complex hyperbolic spaces Hn(C)(n = 4, 6, ... ), Hn-2 (C)

Quaternian hyperbolic spaces Hn(H)(n = 8, 12, ... ), Hn-4 (H)

Cayley hyperbolic spaces H 16 (Cay), H 8 (.R) .

Let 3 denote the set of submanifolds g · eo of X as g runs through G; 3 is given the canonical differentiable structure of a homogeneous space. Each e E 3 has a measure m induced by the Riemannian structure of X and the Radon transform on X is defined by

f(e) = i f(x) dm(x), f E Cc(X).

The dual transform <p --+ if; is defined by

ip(x) = { <p(~) dJ.L(~), <p E C(3), J{3z

§4 The X-ray Transform on a Symmetric Space 119

where p. is the invariant average on the set of~ passing through x. Let L denote the Laplace-Beltrami operator on X, Riemannian structure being that given by the Killing form of _g.

Theorem 3.1. The Radon transform f-+ f is a one-to-one mapping of V(X) into V(2) and, except for the case X= Hn(JR), n even, is inverted by the formula

X= Hn{IR), n odd:

Q(L) = 'Y (L + (n;~)1) ( L + <n;;)3) ... ( L + 1(~~2)). X =Hn(C):

Q(L) = 'Y ( L + ~(~~~~) ( L + ~(~:~)) · · · ( L + ~~:~~j). X= Hn(H):

Q(L) = 'Y ( L + ~(~~~)) ( L + ~(~:~n · · · ( L + ~t:~;n · X= H 16 { Cay):

Q(L) ="f(L+ 194)2 (L+ 1;)2.

The constants 'Yare obtained from the constants c in {90)-(93) by multiplication by the factor Ox which is the volume of the antipodal manifold in the compact space corresponding to X. This factor is explicitly determined for each X in [GGA], Chapter I, §4.

§4 The X-ray Transform on a Symmetric Space

Let X be a complete Riemannian manifold of dimension > 1 in which any two points can be joined by a unique geodesic. The X-ray transform on X assigns to each continuous function f on X the integrals

(99) f('Y) = i f(x) ds(x),

'Y being any complete geodesic in X and ds the element of arc-length. In analogy with the X-ray reconstruction problem on JRn (Ch.l, §7) one can consider the problem of inverting the X-ray transform f --+ f. With d denoting the distance in X and o E X some fixed point we now define two subspaces of C(X). Let

F(X)

J(X) =

{! E C(X): supd(o,x)kif(x)i < oo for each k ~ 0}

{/ E C(X): supekd(o,x)if(x)i < oo for each k ~ 0}.

Because of the triangle inequality these spaces do not depend on the choice of o. We can informally refer to F(X) as the space of continuous rapidly


decreasing functions and to ~(X) as the space of continuous exponentially decreasing functions. We shall now prove the analog of the support theorem (Theorem 2.6, Ch. I, Theorem 1.2, Ch. III) for the X-ray transform on a symmetric space of the noncom pact type. This general analog turns out to be a direct corollary of the Euclidean case and the hyperbolic case, already done.

Corollary 4.1. Let X be a symmetric space of the noncompact type, B any ball in M.

(i) If a function f E ~(X) satisfies

(100) f{f.) = 0 whenever f. n B = 0, f. a geodesic,

then

(101) f(x) = 0 for x ~B.

In particular, the X-ray transform is one-to-one on ~(X).

(ii} If X has rank greater than one statement (i} holds with ~(X) replaced by F(X).

Proof. Let o be the center of B, r its radius, and let 'Y be an arbitrary geodesic in X through o.

Assume first X has rank greater than one. By a standard conjugacy theorem for symmetric spaces 'Y lies in a 2-dimensional, flat, totally geodesic submanifold of X. Using Theorem 2.6, Ch. I on this Euclidean plane we deduce f(x) = 0 if x E "{, d(o, x) > r. Since 'Y is arbitrary (101) follows.

Next suppose X has rank one. Identifying p with the tangent space Xa let a be the tangent line to 'Y· We can then consider the eigenspace decomposition (98). If b C l'a is a line through the origin then S = Exp(a +b) is a totally geodesic submanifold of X (cf. (iv) in the beginning of §2). Being 2-dimensional and not flat, S is necessarily a hyperbolic space. From Theorem 1.2 we therefore conclude f(x) = 0 for x E "f, d(o,x) > r. Again (101) follows since 'Y is arbitrary.

§5 Maximal Tori and Minimal Spheres in Compact Symmetric Spaces

Let u be a compact semisimple Lie algebra, () an involutive automorphism of u with fixed point algebra t. Let U be the simply connected Lie group with Lie algebra u and Int(u) the adjoint group of u. Then() extends to an involutive automorphism of U and Int(u). We denote these extensions also

§5 Maximal Tori and Minimal Spheres in Compact Symmetric Spaces 121

by() and let K and Ke denote the respective fixed point groups under(). The symmetric space Xe = Int(u)/ Ke is called the adjoint space of (u, ()) (Helgason [1978), p. 327), and is covered by X = U / K, this latter space being simply connected since K is automatically connected.

The flat totally geodesic submanifolds of Xe of maximal dimension are permuted transitively by Int(u) according to a classical theorem of Cartan. Let E9 be one such manifold passing through the origin eK9 in X9 and let H9 be the subgroup of Int(u) preserving E9. We then have the pairs of homogeneous spaces

{102) X 9 = Int(u)/K9 , S9 = Int(u)/H9 •

The corresponding Radon transform f -+ f from C(X9) to C(S9) amounts to

(103) [(E)= L f(x) dm(x), E E S9 ,

E being any flat totally geodesic submanifold of X 9 of maximal dimension and dm the volume element. If X9 has rank one, Eisa geodesic and we are in the situation of Corollary 2.3. The transform {103) is often called the fiat Radon transform.

Theorem 5.1. Assume X9 is irreducible. Then the fiat Radon transform is injective.

For a proof see Grinberg [1992). The sectional curvatures of the space X lie in an interval [0, K]. The space

X contains totally geodesic spheres of curvature K and all such spheres S of maximal dimension are conjugate under U (Helgason [1966b]). Fix one such sphere So through the origin eK and let H be the subgroup of U preserving S0 • Then we have another double fibration

X=U/K, S=U/H

and the accompanying Radon transform

[(S) =Is f(x) da(x).

S E S being arbitrary and da being the volume element on S. It is proved by Grinberg [1994] that injectivity holds in many cases al

though the general question is not fully settled.



As mentioned earlier, it was shown by Funk (1916] that a function I on the twosphere, symmetric with respect to the center, can be determined by the integrals of I over the great circles. When I is rotation-invariant (relative to a vertical axis) he gave an explicit inversion formula, essentially (78) in Proposition 1.16.

The Radon transform on hyperbolic and on elliptic spaces corresponding to k-dimensional totally geodesic submanifolds was defined in the author's paper [1959]. Here and in (1990] are proved the inversion formulas in Theorems 1.5, 1.7, 1.10 and 1.11. See also Semyanisty [1961] and Rubin [1998b]. The alternative version in (60) was obtained by Berenstein and Tarabusi [1991] which also deals with the case of H,. in Hn (where the regularization is more complex). Still another interesting variation of Theorem 1.10 (for k = 1, n = 2) is given by Lissianoi and Ponomarev [1997]. By calculating the dual transform ipp(z) they derive from (30) in Chapter II an inversion formula which has a formal analogy to (38) in Chapter II. The underlying reason may be that to each geodesic 'Yin H 2 one can associate a pair of horocycles tangential to lzl = 1 at the endpoints of 'Y having the same distance from o as 'Y·

The support theorem (Theorem 1.2) was proved by the author ([1964], [1980b]) and its consequence, Cor. 4.1, pointed out in [1980d]. Interesting generalizations are contained in Boman [1991], Boman and Quinto [1987], [1993]. For the case of sn-l see Quinto [1983] and in the stronger form of Theorem 1.17, Kurusa (1994]. The variation (82) of the Funk transform has also been considered by Abouelaz and Daher [1993] at least for K-invariant functions. The theory of the Radon transform for antipodal manifolds in compact two-point homogeneous spaces (Theorem 2.2) is from Helgason [1965a]. R. Michel has in [1972] and [1973] used Theorem 2.2 in establishing certain infinitesimal rigidity properties of the canonical metrics on the real and complex projective spaces. See also Guillemin [1976] and A. Besse [1978], Goldschmidt [1990], Estezet (1988].

CHAPTER IV

ORBITAL INTEGRALS AND THE WAVE OPERATOR FOR ISOTROPIC LORENTZ SPACES

In Chapter II, §3 we discussed the problem of determining a function on a

homogeneous space by means of its integrals over generalized spheres. We

shall now solve this problem for the isotropic Lorentz spaces (Theorem 4.1

below). As we shall presently explain these spaces are the Lorentzian analogs of the two-point homogeneous spaces considered in Chapter III.

§ 1 Isotropic Spaces

Let X be a manifold. A pseudo-Riemannian structure of signature (p, q)

is a smooth assignment y -+ gy where y E X and gy is a symmetric non

degenerate bilinear form on Xy x Xy of signature (p, q). This means that

for a suitable basis Y1 , ... , Yp+q of Xy we have

gy(Y) = Yi + · · · + Y;- Y;+1- · · ·- Y;+q

if Y = l:i'+q Yi Yi. H q = 0 we speak of a Riemannian structure and if

p = 1 we speak of a Lorentzian structure. Connected manifolds X with

such structures g are called pseudo-Riemannian (respectively Riemannian,

Lorentzian) manifolds. A manifold X with a pseudo-Riemannian structure g has a differential

operator of particular interest, the so-called Laplace-Beltrami operator. Let

(x1, ... , Xp+q) be a coordinate system on an open subset U of X. We define

the functions 9ii, gii, and g on U by

j

The Laplace-Beltrami operator Lis defined on U by

for f E C00 (U). It is well known that this expression is invariant under

coordinate changes so L is a differential operator on X. An isometry of a pseudo-Riemannian manifold X is a diffeomorphism

preserving g. It is easy to prove that Lis invariant under each isometry <p,

that is L(f o <p) = (Lf) o <p for each f E E(X). Let I(X) denote the group

of all isometries of X. For y E X let I(X)y denote the subgroup of I(X)

124 Chapter IV. Orbital Integrals

fixing y (the isotropy subgroup at y) and let H 11 denote the group of linear transformations of the tangent space X 11 induced by the action of I(X) 11 •

For each a E R let La (y) denote the "sphere"

(1) 'Ea(Y) = {Z E X 11 : g11(Z,Z) =a, Z # 0}.

Definition. The pseudo-Riemannian manifold X is called isotropic if for each a E Rand each y EX the group H11 acts transitively on Ea(y).

Proposition 1.1. An isotropic pseudo-Riemannian manifold X is homogeneous; that is, I(X) acts transitively on X.

Proof The pseudo-Riemannian structure on X gives an affine connection preserved by each isometry g E I(X). Any two points y, z E X can be joined by a curve consisting of finitely many geodesic segments 'Yi (1 :5 i :5 p). Let 9i be an isometry fixing the midpoint of ri and reversing the tangents to ri at this point. The product gp · · · g1 maps y to z, whence the homogeneity of X.

A. The Riemannian Case

The following simple result shows that the isotropic spaces are natural generalizations of the spaces considered in the last chapter.

Proposition 1.2. A Riemannian manifold X is isotropic if and only if it is two-point homogeneous.

Proof If X is two-point homogeneous and y E X the isotropy subgroup I(X)11 at y is transitive on each sphere Sr(Y) in X with center y so X is clearly isotropic. On the other hand if X is isotropic it is homogeneous (Prop. 1.1) hence complete; thus by standard Riemannian geometry any two points in X can be joined by means of a geodesic. Now the isotropy of X implies that for each y E X, r > 0, the group I(X)11 is transitive on the sphere Sr(Y), whence the two-point homogeneity.

B. The General Pseudo-Riemannian Case

Let X be a manifold with pseudo-Riemannian structure g and curvature tensor R. Let y E X and S C X 11 a 2-dimensional subspace on which g11 is nondegenerate. The curvature of X along the section S spanned by Z and Y is defined by

K(S) = 9p(Rp(Z, Y)Z, Y) 9p(Z, Z)gp(Y, Y) - gp(Z, Y)2

The denominator is in fact # 0 and the expression is independent of the choice of Z and Y.

§1 Isotropic Spaces 125

We shall now construct isotropic pseudo-Riemannian manifolds of signature (p, q) and constant curvature. Consider the space JRP+q+l with the flat pseudo-Riemannian structure

Be(Y) = y~ + · · · + y;- Y~1- · · ·- Y;+q + ey~q+l,

Let Q e denote the quadric in JRP+q+l given by

(e = ±1).

Be(Y) =e.

The orthogonal group O(Be) (= O(p,q+ 1) or O(p+ 1, q)) acts transitively on Qe; the isotropy subgroup at o = (0, ... , 0, 1) is identified with O(p, q).

Theorem 1.3. (i) The restriction of Be to the tangent spaces to Qe gives a pseudo-Riemannian structure ge on Qe of signature (p, q).

(ii) We have

(2) Q-1 ~ O(p,q + 1)/0(p,q) (diffeomorphism)

and the pseudo-Riemannian structure g-1 on Q-1 has constant curvature -1.

{iii) We have

{3) QH = O(p+ 1,q)JO(p,q) (diffeomorphism)

and the pseudo-Riemannian structure g+l on Q+l has constant curvature +1.

(iv) The flat space w+q with the quadratic form go(Y) = 'Li YT- 'L~!i YJ and the spaces

O(p, q + 1)/0(p, q), O(p + 1, q)JO(p, q)

are all isotropic and (up to a constant factor on the pseudo-Riemannian structure) exhaust the class of pseudo-Riemannian manifolds of constant curvature and signature (p, q) except for local isometry.

Proof. If s0 denotes the linear transformation

then the mapping u : g -+ s0 gs0 is an involutive automorphism of O(p, q+ 1) whose differential du has fixed point set o(p, q) (the Lie algebra of O(p, q)). The { -1 )-eigenspace of du, say m, is spanned by the vectors

(4)

(5) Yi = Ei,p+q+l + Ep+q+l,i

lj = Ei,p+q+l - Ep+q+l,i

{1:::; i ::;p)' (p + 1 :::; j :::; p + q) .


Here Eij denotes a square matrix with entry 1 where the ith row and the jth column meet, all other entries being 0.

The mapping 1/J : gO(p, q) -t g · o has a differential d'l/J which maps m bijectively onto the tangent plane YP+q+l = 1 to Q-1 at o and d'l/J(X) =X ·o (X Em). Thus

d'l/J(Yk) = (81k, ... , 8p+q+t,k), (1 ~ k ~ p + q).

Thus

proving (i). Next, since the space (2) is symmetric its curvature tensor satisfies

Ra(X, Y)(Z) = [[X, Y], Z],

where [, ] is the Lie bracket. A simple computation then shows for k =f. f.

K(llWk + llUl) = -1 (1 ~ k, f.~ p + q)

and this implies (ii). Part (iii) is proved in the same way. For (iv) we first verify that the spaces listed are isotropic. Since the isotropy action of O(p, q + 1) 0 = O(p, q) on m is the ordinary action of O(p, q) on w+q it suffices to verify that JRP+q with the quadratic form g0 is isotropic. But we know O(p, q) is transitive on Ye = + 1 and on Ye = -1 so it remains to show O(p, q) transitive on the cone {Y =f. 0: Ye(Y) = 0}. By rotation in W' and in JRq it suffices to verify the statement for p = q = 1. But for this case it is obvious. The uniqueness in (iv) follows from the general fact that a symmetric space is determined locally by its pseudo-Riemannian structure and curvature tensor at a point (see e.g. [DS], pp. 200-201). This finishes the proof.

The spaces (2) and (3) are the pseudo-Riemannian analogs of the spaces O(p, 1)/0(p), O(p+ 1)/0(p) from Ch. III, §1. But the other two-point homogeneous spaces listed inCh. III, §2-§3 have similar pseudo-Riemannian analogs (indefinite elliptic and hyperbolic spaces over C, Hand Cay). As proved by Wolf [1967], p. 384, each non-flat isotropic pseudo-Riemannian manifold is locally isometric to one of these models.

We shall later need a lemma about the connectivity of the groups O(p, q). Let Ip,q denote the diagonal matrix ( dij) with

so a matrix g with transpose tg belongs to O(p, q) if and only if

(6)

§1 Isotropic Spaces 127

If y E JRP+q let

yT = (y1 , .•. ,yp,O ... O),y8 = (0, ... ,O,yp+l,··· ,Yp+q)

and for 9 E O{p, q) let 9T and 9s denote the matrices

(9T)ij=9ij (1~i,j~p), (9s)kt=9kl {p+1~k,l~p+q) ·

If 911 ••• ,gp+q denote the column vectors of the matrix 9 then (6) means for the scalar products

1 ~ i ~ p,

p+1~j~p+q,

j of k.

Lemma 1.4. We have for each 9 E O{p, q)

I det(9T)I :?: 1, I det(9s)l :?: 1.

The components of O(p, q) are obtained by

(7) detgT:?: 1 '

det9s:?: 1; (identity component)

(8) detgT ~ -1 ' det9s :?: 1;

(9) detgT:?: -1 ' det9s ~ -1,

{10) det9T ~ -1 '

det9s ~ -1.

Thus O(p, q) has four components if p:?: 1, q :?: 1, two components if p or q=O.

Proof. Consider the Gram determinant

(

T T T T 91 . 9t 91 . 92 ... T T 92 ·g1 .

det

9l·9[

which equals (detgT)2 • Using the relations above it can also be written

(

1 + gf . 9f gf . g~ ... s s 92 . 91 . . ..

det

g: ·9f

which equals 1 plus a sum of lower order Gram determinants each of which is still positive. Thus (detgT)2 :?: 1 and similarly (det98)2 :?: 1. Assuming


now p ~ 1, q ~ 1 consider the decomposition of O{p, q) into the four pieces (7), (8), (9), (10). Each of these is # 0 because (8) is obtained from (7) by multiplication by I1,IJ+q-1 etc. On the other hand, since the functions g -+ det(gT), g-+ det(gs) are continuous on O{p, q) the four pieces above belong to different components of O{p, q). But by Chevalley [1946], p. 201, O{p, q) is homeomorphic to the product of O(p,q) n U{p + q) with a Euclidean space. Since O{p, q) n U{p + q) = O(p, q) n O{p + q) is homeomorphic to O{p) x O(q) it just remains to remark that O(n) has two components.

C. The Lorentzian Case

The isotropic Lorentzian manifolds are more restricted than one might at first think on the basis of the Riemannian case. In fact there is a theorem of Lichnerowicz and Walker [1945] (see Wolf [1967], Ch. 12) which implies that an isotropic Lorentzian manifold has constant curvature. Thus we can deduce the following result from Theorem 1.3.

Theorem 1.5. Let X be an isotropic Lorentzian manifold {signature {1, q), q ~ 1). Then X has constant curvature so (after a multiplication of the Lorentzian structure by a positive constant) X is locally isometric to one of the following:

JRl+q (flat, signature (1, q)),

Q-1 = 0(1,q + 1)/0(1,q): y~- y~- · · ·- Y~+2 = -1,

Q+l = 0(2,q)/0(1,q): y~- y~- · · ·- Y~H + Y~+2 = 1,

the Lorentzian structure being induced by y~ - y~ - · · · =F y~+2 •

§2 Orbital Integrals

The orbital integrals for isotropic Lorentzian manifolds are analogs to the spherical averaging operator Mr considered in Ch. I, §1, and Ch. III, §1. We start with some geometric preparation.

For manifolds X with a Lorentzian structure g we adopt the following customary terminology: If y E X the cone

Cy = {Y E Xy : gy(Y, Y) = 0}

is called the null cone (or the light cone) in Xy with vertex y. A nonzero vector Y E Xy is said to be timelike, isotropic or spacelike if gy (Y, Y) is positive, 0, or negative, respectively. Similar designations apply to geodesics according to the type of their tangent vectors.

While the geodesics in JRl+q are just the straight lines, the geodesics in Q-1 and Q+1 can be found by the method of Ch. ill, §1.

§2 Orbital Integrals 129

Proposition 2.1. The geodesics in the Lorentzian quadrics Q_1 and Q+l have the following properties:

{i) The geodesics are the nonempty intersections of the quadrics with two-planes in JR2+q through the origin.

{ii} For Q-1 the spacelike geodesics are closed, for Q+l the timelike geodesics are closed.

{iii} The isotropic geodesics are certain straight lines in JR2+q .

Proof Part (i) follows by the symmetry considerations in Ch. III, §1. For Part (ii) consider the intersection of Q_1 with the two-plane

Y1 = Y4 = · · · = Yq+2 = 0 .

The intersection is the circle Y2 = cost, Y3 = sin t whose tangent vector (0,- sin t, cost, 0, ... , 0) is clearly spacelike. Since 0(1, q+ 1) permutes the spacelike geodesics transitively the first statement in (ii) follows. For Q+1

we intersect similarly with the two-plane

Y2 = · · · = Yq+l = 0.

For {iii) we note that the two-plane JR{1, 0, ... , 0, 1) + JR(O, 1, ... , 0) intersects Q _1 in a pair of straight lines

Yt = t,y2 ± 1,y3 = · · · = Yq+l = O,yq+2 = t

which clearly are isotropic. The transitivity of 0(1, q + 1) on the set of isotropic geodesics then implies that each of these is a straight line. The argument for Q+l is similar.

Lemma 2.2. The quadrics Q_1 and Q+l (q 2: 1) are connected.

Proof. The q-sphere being connected, the point (Yt, ... , Yq+2) on Q~1 can be moved continuously on Q~1 to the point

(Yt, (y~ + · · · + Y;+1 )112 , 0, . · · , 0, Yq+2)

so the statement follows from the fact that the hyperboloids Yt - YT =t= y~ = =t= 1 are connected.

Lemma 2.3. The identity components of0(1,q+1) and 0(2,q) act transitively on Q_1 and Q+17 respectively, and the isotropy subgroups are connected.


Proof. The first statement comes from the general fact (see e.g [DS], pp. 121-124) that when a separable Lie group acts transitively on a connected manifold then so does its identity component. For the isotropy groups we use the description (7) of the identity component. This shows quickly that

0 0 (1, q + 1) n 0(1, q) = 0 0 (1, q),

Oo(2, q) n 0(1, q) = Oo(1, q)

the subscript o denoting identity component. Thus we have

proving the lemma.

Q-1 = Oo(1,q+1)j00 (1,q),

Q+l = Oo(2,q)/Oo(1,q),

We now write the spaces in Theorem 1.5 in the form X = G j H where H = 0 0 (1, q) and a is either ao = JRl+q · 0 0 (1, q) (semi-direct product) a-= 0 0 (1,q + 1) or a+ = 0 0 (2,q). Let o denote the origin {H} in X, that is

0 = (0, ... ,0) if X= JRl+q

0 = (0, ... '0, 1) if X= Q-1 or Q+1.

In the cases X = Q-1. X = Q+l the tangent space Xo is the hyperplane {Yll · · · , Yq+b 1} C JR2+q.

The timelike vectors at o fill up the "interior" cg of the cone C0 • The set Cg consists of two components. The component which contains the tirnelike vector

V 0 = (-1,0, ... ,0)

will be called the solid retrograde cone in X 0 • It will be denoted by D0 •

The component of the hyperboloid g0 (Y, Y) = r 2 which lies in Do will be denoted Sr ( o). If y is any other point of X we define Cy, Dy, Sr (y) C Xy by

Cy = g ·Co, Dy = 9 ·Do, Sr(Y) = 9 · Sr(o)

if 9 E G is chosen such that 9 · o = y. This is a valid definition because the connectedness of H implies that h ·DoC D0 • We also define

Br(Y) = {Y E Dy: 0 < 9y(Y, Y) < r 2}.

If Exp denotes the exponential mapping of Xy into X, mapping rays through 0 onto geodesics through y we put

Dy = ExpDy,

Sr(Y) = ExpSr(Y),

Cy = ExpCy

Br(Y) = ExpBr(Y).


Exp(rv0) = (-shr, 0, chr)

FIGURE IV.l.

Again Cy and Dy are respectively called the light cone and solid retrograde cone in X with vertex y. For the spaces X= Q+ we always assume r < 1r

in order that Exp will be one-to-one on Br(Y) in view of Prop. 2.1(ii). Figure IV.1 illustrates the situation for Q-1 in the case q = 1. Then Q-1

is the hyperboloid

2 2 2 1 Y1- Y2- Ya =-

and the y1-axis is vertical. The origin o is

0 = (0,0, 1)

and the vector v0 = ( -1, 0, 0) lies in the tangent space

(Q_I)o = {y: Ya = 1}

pointing downward. The mapping 1/J: gH -t g · o has differential d'l/J: m -t (Q-1)o and


in the notation of (4). The geodesic tangent to v 0 at o is

t -4 Exp{tv0 ) = exp( -t(E1a + Eal)) · o = {-sinh t, 0, cosh t)

and this is the section of Q_1 with the plane y2 = 0. Note that since H preserves each plane Ya = const., the "sphere" Sr(o) is the plane section Ya = coshr,y1 < 0 with Q-1·

Lemma 2.4. The negative of the Lorentzian structure on X = G I H induces on each Sr (y) a Riemannian structure of constant negative curvature (q > 1).

Proof. The manifold X being isotropic the group H = 0 0 (1, q) acts transitively on Sr(o). The subgroup leaving fixed the geodesic from o with tangent vector v0 is 0 0 (q). This implies the lemma.

Lemma 2.5. The timelike geodesics from y intersect Sr(Y) under a right angle.

Proof. By the group invariance it suffices to prove this for y = o and the geodesic with tangent vector v0 • For this case the statement is obvious.

Let r(g) denote the translation xH -t gxH on GIH and for Y Em let Ty denote the linear transformation Z -t [Y, [Y, Z)] of m into itself. As usual, we identify m with (GIH) 0 •

Lemma 2.6. The exponential mapping Exp : m -4 G I H has differential

oo Tn dExpy = dr(exp Y) o L (2n ~ 1)!

0

For the proof see [DS), p. 215.

Lemma 2.7. The linear transformation

oo Tn

Ay = ~ (2n ~ 1)!

has determinant given by

{ sinh{g(Y, Y)) 112 }q detAy =

(g(Y, Y))1/2

detAy

for Y timelike.

= {· sin(g(Y, Y)) 112 }q (g(Y, Y) )112

(Y Em).

for Q-1


Proof. Consider the case of Q_1• Since det(Ay) is invariant under H it suffices to verify this for Y = cY1 in (4), where c E JR.. We have c2 = g(Y, Y) and Ty1 (}j) = }j (2 ~ j ~ q + 1). Thus Ty has the eigenvalue 0 and g(Y, Y); the latter is a q-tuple eigenvalue. This implies the formula for the determinant. The case Q+1 is treated in the same way.

From this lemma and the description of the geodesics in Prop. 2.1 we can now conclude the following result.

Proposition 2.8. (i) The mapping Exp: m -t Q_1 is a diffeomorphism of Do onto Do.

{ii) The mapping Exp : m ---* Q +1 gives a diffeomorphism of B1r ( o) onto B1r(o).

Let dh denote a hi-invariant measure on the unimodular group H. Let u E "D(X), y E X and r > 0. Select g E G such that g · o = y and select x E Sr ( o). Consider the integral

[ u(gh · x) dh.

Since the subgroup K C H leaving x fixed is compact it is easy to see that the set

C9 ,x ={hE H: gh · x E support (u)}

is compact; thus the integral above converges. By the bi-invariance of dh it is independent of the choice of g (satisfying g · o = y) and of the choice of x E Sr(o). In analogy with the Riemannian case (Ch. III, §1) we thus define the operator Mr (the orbital integral) by

(11) (Mru)(y) = [ u(gh · x) dh.

If g and x run through suitable compact neighborhoods, the sets C9 ,x are enclosed in a fixed compact subset of H so (Mru)(y) depends smoothly on both r and y. It is also clear from (11) that the operator Mr is invariant under the action of G: if m E G and r(m) denotes the transformation nH -t mnH of G / H onto itself then

If dk denotes the normalized Haar measure on K we have by standard invariant integration

f u(h · x) dh = f dh f u(hk · x) dk = f u(h · x) dh, jH jH/K jK jH/K


where dh is an H-invariant measure on HIK. But if dwr is the volume element on Sr(o) (cf. Lemma 2.4) we have by the uniqueness of H-invariant measures on the space HI K ~ Sr ( o) that

(12) { u(h · x) dh = A1( ) { u(z) dwr(z), JH r ls .. (o)

where A(r) is a positive scalar. But since g is an isometry we deduce from (12) that

No:w we have to determine A(r).

Lemma 2.9. For a suitable fixed normalization of the Haar measure dh on H we have

A(r)=rq, (sinhr)q, (sinr)q

for the cases

JR.l+q, 0(1,q+ 1)10(1,q), 0(2,q)I0(1,q),

respectively.

Proof. The relations above show that dh = A(r)-1 dwr dk. The mapping Exp : D 0 --t D 0 preserves length on the geodesics through o and maps Sr(o) onto Sr(o). Thus if z E Sr(o) and Z denotes the vector from 0 to z in Xo the ratio of the volume of elements of Sr(o) and Sr(o) at z is given by det(dExpz). Because of Lemmas 2.6-2.7 this equals

1, (si:hr) q, (si~rr

for the three respective cases. But the volume element dwr on Sr(o) equals rqdw1 • Thus we can write in the three respective cases

rq • hq . q dh dw1 dk sm r dw1 dk sm r dw1 dk

= A(r) ' A(r) ' A(r) ·

But we can once and for all normalize dh by dh = dw1 dk and for this choice our formulas for A(r) hold.

Let 0 denote the wave operator on X = G I H, that is the LaplaceBeltrami operator for the Lorentzian structure g.

Lemma 2.10. Let y EX. On the solid retrograde cone Dy, the wave operator 0 can be written

82 1 dA8 0 = 8r2 + A(r) dr 8r- Ls .. (v)'

where Ls .. (y) is the Laplace-Beltrami operator on Sr(y).


Proof. We can take y = o. If ( 81, ... , 8 q) are coordinates on the "sphere" S1(o) in the fiat space Xo then (rfh, ... , r8q) are coordinates on Sr(o). The Lorentzian structure on Do is therefore given by

dr2 - r 2 d82 ,

where d82 is the Riemannian structure of 81 (o). Since Ay in Lemma 2.7 is a diagonal matrix with eigenvalues 1 and r-1 A(r)1fq (q-times) it follows from Lemma 2.6 that the image Sr(o) = Exp(Sr(o)) has Riemannian structure r2 d(J2, sinh 2 r d82 and sin 2 r d82 in the cases JR.l+q , Q -1 and Q +1 , respectively. By the perpendicularity in Lemma 2.5 it follows that the Lorentzian structure on Do is given by

in the three respective cases. Now the lemma follows immediately.

The operator Mr is of course the Lorentzian analog to the spherical mean value operator for isotropic Riemannian manifolds. We shall now prove that in analogy to the Riemannian case ( cf. ( 41), Ch. III) the operator Mr commutes with the wave operator D.

Theorem 2.11. For each of the isotropic Lorentz spaces X = a-IH, a+ I H or ao I H the wave operator 0 and the orbital integral Mr commute:

for u E V(X).

(For a+ IH we assume r < 1r.)

Given a function u on a1H we define the function u on a by u(g) = u(g · o).

Lemma 2.12. There exists a differential operator 0 on a invariant under all left and all right translations such that

Du = (Du)~ for u E V(X).

Proof. We consider first the case X = a- I H. The bilinear form

K(Y,Z) = !Tr(YZ)

on the Lie algebra o(1, q + 1) of a- is nondegenerate; in fact K is nondegenerate on the complexification o(q + 2, C) consisting of all complex skew symmetric matrices of order q + 2. A simple computation shows that in the notation of (4} and (5)

K(Y1, Y1) = 1, K(}j, lj) = -1 (2 s; j s; q + 1).

Since K is symmetric and nondegenerate there exists a unique left invariant pseudo-Riemannian structure K on a- such that Ke = K. Moreover, since


K is invariant under the conjugation Y-+ gY g-1 of o(l, q + 1), K is also right invariant. Let 0 denote the corresponding Laplace-Beltrami operator on a-. Then 0 is invariant under all left and right translations on a-. Let u = V(X). Since Ou is invariant under all right translations from H there is a unique function v E £(X) such that Ou = v. The mapping u-+ v is a differential operator which at the origin must coincide with 0, that is Ou(e) = Ou(o). Since, in addition, both 0 and the operator u -+ v are invariant under the action of a- on X it follows that they coincide. This proves Ou = (Ou )"'.

The case X = a+/ H is handled in the same manner. For the flat case X= G0 /H let

}j = (0, ... ' 1, ... '0) '

the jth coordinate vector on JRl+q. Then 0 = Y[ - Y22 - • • • - Yq;_1 • Since JRl+q is naturally embedded in the Lie algebra of G0 we can extend }j to a left invariant vector field lJ on G0 • The operator

- -2 -2 -2 0 = yl - y2 - ... - Yq+l

is then a left and right invariant differential operator on G0 and again we have Ou = (Ou)"'. This proves the lemma.

We can now prove Theorem 2.11. If g E G let L(g) and R(g), respectively, denote the left and right translations £ -+ g£, and£ -+ ig on G. If£· o = x,x E Sr(o) (r > 0) and g · o = y then

(Mru)(y) = [ u(gh£) dh

because of (11). As g and£ run through sufficiently small compact neighborhoods the integration takes place within a fixed compact subset of H as remarked earlier. Denoting by subscript the argument on which a differential operator is to act we shall prove the following result.

Lemma 2.13.

Eit (i u(gh£) dh) = L (Du)(gh£) dh = 159 (i u(gh£) dh)

Proof. The first equality sign follows from the left invariance of D. In fact, the integral on the left is

so

[ (u o L(gh))(£) dh

Ell ([ u(ght) dh) = [ [o<u o L(gh))] (£) dh

= [ [<ElU) o L(gh)] (£) dh = [ (Du)(gh£) dh.

§3 Generalized Riesz Potentials 137

The second equality in the lemma follows similarly from the right invariance of 0. But this second equality is just the commutativity statement in Theorem 2.11.

Lemma 2.13 also implies the following analog of the Darboux equation in Lemma 3.2, Ch. I.

Corollary 2.14. Let u E 'D(X) and put

U(y, z) = (Mru)(y) if z E Sr(o).

Then Dy(U(y, z)) = Dz(U(y, z)).

Remark 2.15. In R.n the solutions to the Laplace equation Lu = 0 are characterized by the spherical mean-value theorem Mru = u (all r). This can be stated equivalently: Mr u is a constant in r. In this latter form the mean value theorem holds for the solutions of the wave equation Ou = 0 in an isotropic Lorentzian manifold: If u satisfies Ou = 0 and if u is suitably small at oo then ( Mr u) ( o) is constant in r. For a precise statement and proof see Helgason [1959], p. 289. For R.2 such a result had also been noted by Asgeirsson.

§3 Generalized Riesz Potentials

In this section we generalize part of the theory of Riesz potentials ( Ch. V, §5) to isotropic Lorentz spaces.

Consider first the case

X= Q-1 = G-/H = 0 0 (1,n)/Oo(1,n-1)

of dimension nand let f E 'D(X) andy EX. If z = ExpyY (Y E Dy) we put ryz = g(Y, Y) 112 and consider the integral

{13) (I~f)(y) = Hn~A) L11

f(z) sinh.X-n(ryz) dz,

where dz is the volume element on X, and

(14) Hn(>-.) = 7r(n-2)/22.x-1r (>-./2) r ((>-. + 2- n)/2) .

The integral converges for Re).. ~ n. We transfer the integral in (13) over to Dy via the diffeomorphism Exp(= Expy). Since

( · h )n-1

dz = drdwr = dr su:. r ru,.;r


and since drdJ...Jr equals the volume element dZ on Dy we obtain

1 r (sinhr)>.-1 (I>.f)(y)= Hn(A)}nv(foExp)(Z) -r- r>.-naz,

where r = g(Z, Z) 112 • This has the form

(15) Hn1(A) Lv h(Z, A)r>.-n dZ'

where h(Z, A), as well as each of its partial derivatives with respect to the first argument, is holomorphic in A and h has compact support in the first variable. The methods of Riesz [1949], Ch. III, can be applied to such integrals (15). In particular we find that the function A -+ (I~f)(y) which by its definition is holomorphic for ReA > n admits a holomorphic continuation to the entire A-plane and that its value at A = 0 is h(O, 0) = f(y). (In Riesz' treatment h(Z, A) is independent of A, but his method still applies.) Denoting the holomorphic continuation of {13) by (I~)f(y) we have thus obtained

(16) I?_j =f.

We would now like to differentiate (13) with respect toy. For this we write the integral in the form JF f(z)K(y, z) dz over a bounded region F which properly contains the intersection of the support of f with the closure of Dy. The kernel K(y, z) is defined as sinh>.-n ryz if z E D 11 , otherwise 0. For ReA sufficiently large, K (y, z) is twice continuously differentiable in y so we can deduce for such A that I~ f is of class C2 and that

Moreover, given m E Z + we can find k such that I~ f E em for ReA > k (and all f). Using Lemma 2.10 and the relation

1 dA A(r) dr = (n- 1) coth r

we find

0 11 (sinh>.-n r11z) = Dz(sinh>.-n r11z)

= (A-n)(A-1) sinh>.-n ryz+(A-n)(A-2) sinh>.-n-2 ryz.

We also have

Hn(A) = (A- 2)(A- n)Hn(A- 2)

so substituting into (17) we get

§3 Generalized Riesz Potentials 139

OI~f = (A- n)(A- 1)I~f + I~-2 f.

Still assuming ReA large we can use Green's formula to express the integral

(18) { [f(z)Oz(sinh>.-n ryz)- sinh>.-n ryz(Of)(z)] dz loy

as a surface integral over a part of Cy (on which sinh>.-n ryz and its first order derivatives vanish) together with an integral over a surface inside Dy (on which f and its derivatives vanish). Hence the expression (18) vanishes so we have proved the relations

(19)

(20)

O(I~f) = J~(Of)

I~(Of) = (A- n)(A- 1)I~f + I~-2 f

for ReA> k, k being some number (independent of f). Since both sides of (20) are holomorphic in A this relation holds for all

A E C. We shall now deduce that for each A E C, we have I~f E £(X) and (19) holds. For this we observe by iterating (20) that for each p E z+

(21)

Qp being a certain pth_degree polynomial. Choosing p arbitrarily large we deduce from the remark following (17) that I~f E £(X); secondly (19) implies for Re A + 2p > k that

OI~+2P(Qp(O)f) = I~+2P(Qp(O)Of).

Using (21) again this means that (19) holds for all A. Putting A= 0 in (20) we get

(22) T:.2 = Of - nf.

Remark 3.1. In Riesz' paper [1949), p. 190, an analog F" of the potentials in Ch. V, §5, is defined for any analytic Lorentzian manifold. These potentials JOt are however different from our I~ and satisfy the equation r- 2 f = Of in contrast to (22).

We consider next the case

X= Q+l =a+ /H = 0 0 (2,n-1)/0o(1,n-1)

and we define for f E V(X)

(23) (I~f)(y) = Hn1(A) LY f(z) sin>.-n(ryz) dz.


Again Hn(A) is given by (14) and dz is the volume element. In order to bypass the difficulties caused by the fact that the function z -t sin r11z vanishes on 811' we assume that f has support disjoint from 811' ( o). Then the support off is disjoint from 81r(y) for ally in some neighborhood of o in X. We can then prove just as before that

(24) <4-f)(y) = f(y)

(25) (OI~f)(y) = (I~Of)(y)

(26) (1~0/)(y) = -(A- n)(A- 1)(1~/)(y) + (I~-2 f)(y)

for all A E C. In particular

(27) If.2 I = Of + nf.

Finally we consider the flat case

X= r = 00 /H = r · 0 0 (1,n-1)/00 (1,n-1)

and define

(I; f)(y) = Hn1(A) L11

f(z)r;;n dz.

These are the potentials defined by Riesz in [1949], p. 31, who proved

(28) I~!=/, 01; I= I:ot = 1;-2 I.

§4 Determination of a Function from its Integral over Lorentzian Spheres

In a Riemannian manifold a function is determined in terms of its spherical mean values by the simple relation f = limr-to Mr f. We shall now solve the analogous problem for an even-dimensional isotropic Lorentzian manifold and express a function f in terms of its orbital integrals Mr f. Since the spheres Sr(Y) do not shrink to a point as r -t 0 the formula (cf. Theorem 4.1) below is quite different.

For the solution of the problem we use the geometric description of the wave operator 0 developed in §2, particularly its commutation with the orbital integral Mr, and combine this with the results about the generalized Riesz potentials established in §3.

We consider first the negatively curved space X = a-/H. Let n = dim X and assume n even. Let f E V(X), y EX and put F(r) = (Mr f)(y). Since the volume element dz on D 11 is given by dz = dr dwr we obtain from {12) and Lemma 2.9 ,

(29) (I:f)(y) = Hn1(A) 100 sinh>.-l rF(r) dr.

§4 Determination of a Function from its Integral over Lorentzian Spheres 141

Let Y1, ... , Yn be a basis of Xy such that the Lorentzian structure is given by

n

9y(Y) = yf - y~ - ... - y~ ' y = L YiYi . 1

If 81, ... , Bn-2 are geodesic polar coordinates on the unit sphere in JRn-1 we put

Y1 = -rcosh( (0 :::; ( < oo, 0 < r < oo)

Y2 = r sinh (cos fh

Yn = r sinh (sin fh ... sin Bn-2 .

Then (r, (, 01 , •.. , Bn-2 ) are coordinates on the retrograde cone Dy and the volume element on Sr(Y) is given by

dwr = rn-1 sinhn-2 ( d( dwn-2

where dwn-2 is the volume element on the unit sphere in JRn-1 • It follows that

and therefore

where for simplicity

stands for

( -r cosh(, r sinh (cos 01 , ... , r sinh (sin 81 ... sin Bn-2).

Now select A such that foExp vanishes outside the sphere yr+· · ·+y~ = A2

in Xy. Then, in the integral {30), the range of (is contained in the interval (0, (0 ) where

Then

rn-2 F(r) = { f'o (f o Exp)(r, (, (O))(r sinh ()n-2 d( dwn-2 • }gn-2 Jo


Since

lrsinh(l:::; re':::; 2A for (:::; ( 0

this implies

(31)

where Cis a constant independent of r. Also substituting t = rsinh( in the integral above, the (-integral becomes

1k cp(t)tn-2(r2 + t2)-1/2 dt'

where k = [(A2 - r 2)/2]112 and cp is bounded. Thus if n > 2 the limit

(32) a= lim sinhn-2 rF(r) n > 2 r-+0

exist and is ~ 0. Similarly, we find for n = 2 that the limit

(33) b = lim(sinhr)F'(r) r-+0

(n = 2)

exists. Consider now the case n > 2. We can rewrite (29) in the form

(I:J)(y) = Hn1(A) 1A sinhn-2 rF(r) sinh.A-n+l rdr,

where F(A) = 0. We now evaluate both sides for A = n- 2. Since Hn(A) has a simple pole for A = n - 2 the integral has at most a simple pole there and the residue is

lim (A- n+2) 1A sinhn-2 rF(r) sinh.A-n+l rdr . .A-+n-2 0

Here we can take A real and greater than n - 2. This is convenient since by (32) the integral is then absolutely convergent and we do not have to think of it as an implicitly given holomorphic extension. We split the integral in two parts

(A-n+2) fo\sinhn-2 rF(r)- a) sinh.A-nt-1 rdr

+a( A-n+ 2) 1A sinh.A-n+l r dr.

For the last term we use the relation

1A 1sinhA lim f-t sinh~'- 1 r dr = lim J.t t~'- 1 (1 + t2 )-112 dt = 1

JL-+0+ 0 JL-+0+ 0

§4 Determination of a Function from its Integral over Lorentzian Spheres 143

by (38) in Chapter V. For the first term we can for each € > 0 find a 8 > 0 such that

I sinhn-2 rF(r)- al < € for 0 < r < 8.

If N =max I sinhn-2 rF(r)l we have for n- 2 <.A< n- 1 the estimate

!(.X-n+2) [A (sinhn-2 rF(r)- a) sinh.\-n+-1 r drl

::::; (.A-n+2)(N + lai)(A- 8}(sinh8)A-n+-1 ;

!(.A+n-2) 16 (sinhn-2 rF(r)- a) sinh.\-n+-1 r dri

::::; €(.A - n + 2) 16 rA-n+l dr = €8A-n+2 .

Taking .A- (n- 2) small enough the right hand side of each of these inequalities is < 2€. We have therefore proved

lim (.A-n+ 2) roo sinhA-1 rF(r) dr = lim sinhn-2 r F(r). A~n-2 }0 r~O

Taking into account the formula for Hn(.A) we have proved for the integral (29):

(34) rn-2/ = (4 )(2-n)/2 1 li · hn-2 Mrf 1_ 1r r((n- 2)/2) r~ sm r .

On the other hand, using formula (20) recursively we obtain for u E V(X)

where

Q(O) = (0 + (n- 3)2)(0 + (n- 5)4) · · · (0 + 1(n- 2)).

We combine this with (34) and use the commutativity OMr = Mro. This gives

(35) u = (47r)<2-n)/2 1 lim sinhn-2 r Q(O)Mru. r((n- 2)/2) r~O

Here we can for simplicity replace sinh r by r. For the case n = 2 we have by (29)

(36) 1 roo

(I'!_f)(y) = H2(2) Jo sinhrF(r) dr.


This integral which in effect only goes from 0 to A is absolutely convergent because our estimate (31) shows (for n = 2) that rF(r) is bounded near r = 0. But using (20), Lemma 2.10, Theorem 2.11 and Cor. 2.14, we obtain for u E 1J(X)

u = 1:ou =! 100 sinhrMrOudr

= ! 100 sinhrDMrudr =! 100

sinhr (!2 + cothr!) urudr

1 100 d ( inh d Mr ) d - 1 lim inh d( Mr U) = - - s r- u r - -- s r . 2 0 dr dr 2 r-+0 dr

This is the substitute for (35) in the case n = 2. The spaces G+ I H and G0 I H can be treated in the same manner. We

have thus proved the following principal result of this chapter.

Theorem 4.1. Let X be one of the isotropic Lorentzian manifolds a- IH, G0 fH, G+ fH. LetK. denote the curoature of X (~t = -1,0, +1) and assume n = dim X to be even, n = 2m. Put

Q(D) = (D- ~t(n- 3)2)(0- ~t(n- 5)4) · · · (0- d(n- 2)).

Then if u E 1J{X)

u = c lim rn-2Q(D)(Mru), (n =f. 2) r-+0

u = -21 lim r.!!_(Mru) (n = 2). r-+0 dr

Here c-1 = (4n')m-1(m-2)! and 0 is the Laplace-Beltrami operator on X.

§5 Orbital Integrals and Huygens' Principle

We shall now write out the limit in (35) and thereby derive a statement concerning Huygens' principle for D. As r --t 0, Sr(o) has as limit the boundary OR= 8D0 - {o} which is still an H-orbit. The limit

{37) v E Oc(X- o)

is by (31)-{32) a positive H-invariant functional with support in the Horbit OR, which is closed in X- o. Thus the limit {37) only depends on the restriction v!OR. Hence it is "the" H-invariant measure on OR and we denote it by p. Thus

(38) lim rn-2 (Mrv)(o) = f v(z) dp(z). r-+0 loR


To extend this to u E 'D(X), let A > 0 be arbitrary and let <p be a "smoothed out" characteristic function of Exp B A. Then if

U1 = U<p, U2 = u(1 - <p)

we have

lrn- 2(Mru)(o)- [R u(z) dJL(z)l

:$ lrn-2(Mrul)(o)-[Rul (z) dJL(z)l + lrn-2(Mru2)(o)-[Ru2(z) dJL(z) I· By (31) the first term on the right is O(A) uniformly in r and by (38) the second tends to 0 as r -+ 0. Since A is arbitrary (38) holds for u E 'D(X).

Corollary 5.1. Let n = 2m (m > 1) and 8 the delta distribution at o. Then

{39) 8 = cQ(O)JL,

where c-1 = (41l')m-l(m- 2)!.

In fact, by (35), {38) and Theorem 2.11

u = c lim rn-2(MrQ(D)u)(o) = c { (Q(D)u)(z) dJL(z) r--+0 JcR

and this is {39).

Remark 5.2. Formula (39) shows that each factor

(40) Dk=O-x;(n-k)(k-1) k=3,5, ... ,n-1

in Q(D) has fundamental solution supported on the retrograde conical surface C R· This is known to be the equivalent to the validity of Huygens' principle for the Cauchy problem for the equation Oku = 0 (see Gunther [1991] and [1988], Ch. IV, Cor. 1.13). For a recent survey on Huygens' principle see Berest [1998].


§1. The construction of the constant curvature spaces (Theorems 1.3 and 1.5) was given by the author ([1959], {1961]). The proof of Lemma 1.4 on the connectivity is adapted from Boerner [1955]. For more information on isotropic manifolds (there is more than one definition) see Tits [1955], p. 183 and Wolf [1967].

§§2-4. This material is based on Ch. IV in Helgason [1959]. Corollary 5.1 with a different proof and the subsequent remark were shown to me by Schlichtkrull.


See Schimming and Schlichtkrull [1994) (in particular Lemma 6.2) where it is also shown that the constants Ck = -K-(n- k)(k- 1) in (40) are the only ones for which 0 + Ck satisfies Huygens' principle. Here it is of interest to recall that in the flat Lorentzian case R2m, 0 + c satisfies Huygens' principle only for c = 0. Theorem 4.1 was extended to pseudo-Riemannian manifolds of constant curvature by Orloff [1985), [1987). For recent representative work on orbital integrals see e.g. Bouaziz [1995], Flicker [1996], Harinck [1998), Renard [1997].

CHAPTER V

FOURIER TRANSFORMS AND DISTRIBUTIONS. A RAPID COURSE

§1 The Topology of the Spaces 'D(R.n), t'(R.n) and S(R.n)

Let lRn = { x = (xt, ... , Xn) : Xi E JR} and let fA denote o / OXi. If (all ... , an) is ann-tuple of integers ai ;::: 0 we put a! = a1! ···an!,

Da. = 8f1 ... 0~" ' xa. = xrl ... X~" ' Ia I = al + ... + an .

For a complex number c, Re c and Im c denote respectively, the real part and the imaginary part of c. For a given compact set K C ~ let

where supp stands for support. The space 1JK is topologized by the seminorms

(1) II!IIK,m = L sup I(Da. f)(x))l, mE z+. la.I:Sm xEK

The topology of 1) = 1J(JRn) is defined as the largest locally convex topology for which all the embedding maps 1JK -t 1) are continuous. This is the so-called inductive limit topology. More explicitly, this topology is characterized as follows:

A convex set C C 1) is a neighborhood of 0 in 1J if and only if for each compact set K c JRn, C n 1JK is a neighborhood of 0 in 1JK.

A fundamental system of neighborhoods in 1) can be characterized by the following theorem. If BR denotes the balllxl < R in JRn then

(2)

Theorem 1.1. Given two monotone sequences

{N} = No,Nt,Nz, ... , Ni-t oo Ni E z+ let V ( { €}, { N}) denote the set of functions <p E 1) satisfying for each j the conditions

(3)

Then the sets V ( { €}, { N}) form a fundamental system of neighborhoods of 0 in 1J.

148 Chapter V. Fourier Transforms and Distributions. A Rapid Course

Proof. It is obvious that each V({e}, {N}) intersects each 'DK in a neighborhood of 0 in 'DK. Conversely, let W be a convex subset of V intersecting each VK in a neighborhood of 0. For each j E z+, 3Nj E z+ and 1Ji > 0 such that each <p E V satisfying

IDa<p(x)l:::; 1Ji for lal:::; Nj supp(<p) C Bj+2

belongs to W. Fix a sequence (/3j) with

/3i E V, /3i ~ 0, 'E/3j = 1, supp(/3j) C Bi+2 - Bi

and write for <p E V,

Then by the convexity of W, <p E W if each function 2i+l /3j <p belongs to W. However, Da (/3i'-P) is a finite linear combination of derivatives Dl3 /3i and fl'Y<p, (1/31, h'l :::; lal). Since (/3j) is fixed and only values of <pin Bj+2- Bj enter, 3 constant kj such that the condition

implies

12j+1 Da(/3j<p)(x)l :::; kj€j for lal :::; Nj, all x.

Choosing the sequence { e} such that kiei :::; 1]j for all j we deduce for each j

<p E V({e}, {N}) => 2i+lf3i'-P E W,

whence <p E W. The space£= &(lRn) is topologized by the seminorms (1) for the vary

ing K. Thus the sets

VJ,k,l = {'-P E &(JR.n): 11'-PIIB";,k < 1/l j, k,l E z+

form a fundamental system of neighborhoods of 0 in £ (JR.n). This system being countable the topology of £ (JR.n) is defined by sequences: A point <p E £ (JR.n) belongs to the closure of a subset A C £ (JR.n) if and only if <p is the limit of a sequence in A. It is important to realize that this fails for the topology of V(JR.n) since the family of sets V ( { e}, { N}) is uncountable.

The space S = S (JR.n) of rapidly decreasing functions on JR.n is topologized by the seminorms (6), Ch. I. We can restrict the Pin (6), Ch. I to polynomials with rational coefficients.

In contrast to the space V the spaces 'DK, £ and S are Fhkhet spaces, that is their topologies are given by a countable family of seminorms.

The spaces 'DK(M), V(M) and &(M) can be topologized similarly if M is a manifold.

§2 Distributions 149

§2 Distributions

A distribution by definition is a member of the dual space V' (JR'l) of 'D(lRn). By the definition of the topology of V, T E 'D' if and only if the restriction TI'Dx is continuous for each compact set K C m.n. Each locally integrable function F on m.n gives rise to a distribution <p--; J <p(x)F(x) dx. A measure on m.n is also a distribution.

The derivative 8iT of a distribution T is by definition the distribution <p --; -T(8i<p). IfF E C1 (1Rn) then the distributions Ta.F and 8i(Tp) coincide (integration by parts).

A tempered distribution by definition is a member of the dual space S' (lRn). Since the imbedding V --; S is continuous the restriction of aT E S' to 'D is a distribution; since V is dense in S two tempered distributions coincide if they coincide on V. In this sense we have S' C 'D'.

Since distributions generalize measures it is sometimes convenient to write

T(<p) = J <p(x) dT(x)

for the value of a distribution on the function <p. A distribution T is said to be 0 on an open set U C m.n if T(<p) = 0 for each <p E V with support contained in U. Let U be the union of all open sets U a C m.n on which T is 0. Then T = 0 on U. In fact, if f E 'D(U), supp(f) can be covered by finitely many Ua, say U1, ... , Ur. Then U1, ... , Ur, JR1l - supp(f) is a covering of m.n. If 1 = :E~+l <pi is a corresponding partition of unity we have f =I:~ <pif so T(f) = 0. The complement m.n- U is called the support of T, denoted supp(T).

A distribution T of compact support extends to a unique element of £' (lRn) by putting

if <po is any function in 'D which is identically 1 on a neighborhood of supp(T). Since 'Dis dense in£, this extension is unique. On the other hand letT E £'(R.n), Tits restriction to 'D. Then supp(T) is compact. Otherwise we could for each j find <pj E £such that <pj = 0 on Bj but T(<pJ) = 1. Then <pj --; 0 in £, yet r( <pj) = 1 which is a contradiction.

This identifies £' (lRn) with the space of distributions of compact support and we have the following canonical inclusions:

V(R.n) C S(R.n) C £(R.n) n n n

£' (JRn) C S' (JRn) C 'D' (JRn) .


If S and T are two distributions, at least one of compact support, their convolution is the distribution S * T defined by

(4) tp -t I tp(x + y) dS(x) dT(y), tp E V(Rn).

If f E V the distribution T1 * T has the form T9 where

g(x) =I f(x- y) dT(y)

so we write for simplicity g = f * T. Note that g(x) = 0 unless x - y E supp(f) for some y E supp(T). Thus supp(g) c supp(f) + suppT. More generally,

supp{S * T) c supp(S) + supp T

as one sees from the special case S = T9 by approximating S by functions S * 1/)E with supp(tpE) C BE(O).

The convolution can be defined for more general S and T, for example if S E S, T E S' then S * T E S'. Also S E £', T E S' implies S * T E S'.

§3 The Fourier Transform

For f E L1 (Rn) the Fourier transform is defined by

(5)

Iff has compact support we can take~ E en. For f E S(Rn) one proves quickly

and this implies easily the following result.

Theorem 3.1. The Fourier transform is a linear homeomorphism of S onto S.

The function 1/J(x) = e-z2 12 on lR satisfies 1/J'{x) +x'I/J = 0. It follows from (6) that ;f satisfies the same differential equation and thus is a constant multiple of e-e/2 • Since ;f(O) = J e_z22 dx = (27r) 112 we deduce ;f(~) = (27r)112e-e2 12 • More generally, if 1/J{x) = e-izi2 12 , (x E JRn) then by product integration

{7)

§3 The Fourier Transform 151

Theorem 3.2. The Fourier transform has the following properties.

(i} f(x) = (2n)-n J i(~)ei(x,~) dt;, for f E S.

(ii} f -t f extends to a bijection of L2 (JR.n) onto itself and

(iii} (h *h)~= Afz for JI,h E S.

(iv} (hh)~ = (2n)-n A* fz for h, f2 E S.

Proof. {i) The integral on the right equals

but here we cannot exchange the integrations. Instead we consider for g E S the integral

which equals the expressions

(8) I J(e)g(~)ei(x,~) d~ =I f(y)g(y- x) dy =I f(x + y)g(y) dy.

Replace g(~) by g(t:.~) whose Fourier transform is cng(y/t:.). Then we obtain

which upon letting € -t 0 gives

g(O) I f(~)ei(x,~) d~ = f(x) I g(y) dy.

Taking g(~) as e-1~12 12 and using (7) Part (i) follows. The identity in (ii) follows from (8) (for x = 0) and (i). It implies that the image L2 (JR.n)~ is closed in L2 (JR.n ). Since it contains the dense subspace S(JR.n) (ii) follows. Formula (iii) is an elementary computation and now (iv) follows taking (i) into account.

If T E S' (JR.1!.) its Fourier transform is the linear form T on S (JR.n) defined by

(9) T(rp) = T((j)).


Then by Theorem 3.1, T E S'. Note that

(10) I cp(~)f(~) df. =I fP(x)f(x) dx

(11)

so the definition (9) extends the old one (5). If 81,82 E £' (JR.n) then 81 and 82 have the form T81 and T82 where 81, 82 E £ (JR.n) and in addition (S1 * S2)"" = Ts182 • We express this in the form

(12)

This formula holds also in the cases

81 E S(JR.n), 82 E S'(JR.n),

81 E £' (JR.n), 82 E S' (JR.n)

and 81 * S2 E S'(JR.n) (cf. Schwartz [1966], p. 268). The classical Paley-Wiener theorem gave an intrinsic description of

L2 (0, 2n")"". We now prove an extension to a characterization of D(JR.n )'"" and £' (JR.n )~.

Theorem 3.3. (i) A holomorphic function F( () on en is the Fourier transform of a distribution with support in B R if and only if for some constants C and N ~ 0 we have

(13)

{ii} F(() is the Fourier transform of a function in VBR (JR.n) if and only if for each N E Z + there exists a constant C N such that

(14)

Proof. First we prove that (13) is necessary. Let T E £' have support in B R

and let x E 1) have support in B R+l and be identically 1 in a neighborhood of BR. Since £(JR.n) is topologized by the semi-norms (1) for varying K and m we have for some Co~ 0 and NEz+

IT(cp)l = IT(xcp)l ~Co L sup l(Da(xcp))(x)!. jaj$N xEBR+l

Computing Da(xcp) we see that for another constant C1


Let 'lj; E £(JR) such that 'lj; = 1 on (-oo,~), and= 0 on (1,oo). Then if ( # 0 the function

cp<;(x) = e-i(z,{)'1f;(!(!(lxl- R))

belongs to V and equals e-i(z,{} in a neighborhood of BR. Hence

(16) T(() = T(cp<) ~ Ct L sup IDacp,j. lai$N

Now supp(cp<) C BR+I<I-t and on this ball

je-i(z,()l ~ elziiim{l ~ e(R+IW 1)1Im(l ~ eRIIm(l+l.

Estimating Dacp, similarly we see that by (16), T(() satisfies (13). The necessity of (14) is an easy consequence of (6). Next we prove the sufficiency of (14). Let

(17)

Because of (14) we can shift the integration in (17) to the complex domain so that for any fixed 17 E JRn ,

f(x) = (2rr)-n { F(~ + i17)ei(z,Hi11} ~. JR."

We use (14) for N = n + 1 to estimate this integral and this gives

1/(x)l ~ CNeRI1Ii-(z,1j}(2rr)-n r (1 + l~lt+l d~. }R.n

Taking now 11 = tx and letting t-+ +oo we deduce f(x) = 0 for !x! > R. For the sufficiency of (13) we note first that F as a distribution on JRn is

tempered. Thus F = J for some f E S'(JRn). Convolving f with a cp E 'DJJ. we see that f * cp satisfies estimates (14) with R replaced by R + c. Thus supp(f * cpe) C BR+e· Letting €-+ 0 we deduce supp(f) C BR, concluding the proof.

We shall now prove a re.§nement of Theorem 3.3 in that the topology of 1J is described in terms of1J. This has important applications to differential equations as we shall see in the next section.

Theorem 3.4. A convex set V C V is a neighborhood of 0 in V if and only if there exist sequences

such that V contains all u E 1) satisfying

(18) 1-(r)j < ~ 0 1 k!Im(l r E en. u ., - 6_ k (1 + l(i)M" e ' .,


The proof is an elaboration of that of Theorem 3.3. Instead of the contour shift Rn -t Rn + i17 used there one now shifts Jr to a contour on which the two factors on the right in (14) are comparable.

Let W({cS}, {M}) denote the set of u E 1J satisfying (18). Given k the set

W~c = {u E '%,. : lu(()l ~ cS~c(1 + 1(1)-M,.ekiim(l}

is contained in W({cS},{M}). Thus if V is a convex set containing W({cS}, {M}) then V n '%,. contains W~c which is a neighborhood of 0 in '%,.(because the bounds on u correspond to the bounds on the lluiiB',.,M,.). Thus Vis a neighborhood of 0 in 1J.

Proving the converse amounts to proving that given V ( { e}, { N}) in Theorem 1.1 there exist {cS}, {M} such that

W( {cS}, {M}) c V({e}, {N}).

For this we shift the contour in (17) to others where the two factors in (14) are comparable. Let

Then

X= (xl! ... ,xn),

(= ((1,•·• ,(n)

( = ~ +if'J,

x' = (x1,··· ,Xn-d (' = ((1, · · · , (n-,1)

~,f'J ERn.

In the last integral we shift from R to the contour in C given by

(20)

m E Z + being fixed. We claim that (cf. Fig. V.1)

(21) r ei:r:n{nu(~', ~n) den= 1 ei:r:n(nu(~', (n) d(n. }R 'Ym

Since (14) holds for each N, u decays between ~n-axis and 'Ym faster than any l(ni-M· Also

ld(nl I . 1 8(1~1), den = 1 + 'm 1 + 1~1 . O~n ~ 1 + m ·

Thus (21) follows from Cauchy's theorem in one variable. Putting

r m = { ( E en : (' E Rn-1 , (n E 'Ym}


m log(l + 1~'1)

FIGURE V.l.

we thus have with d( = d~1 ... d~n-1 d(n,

(22) u(x) = (2n)-n 1 u(()ei(x,(} d(. r,.

Now suppose the sequences {€}, {N} (as in Theorem 1.1) are given. We shall then prove that if 8r, ... , 8j-1 and M1, ... , Mj-1 are already chosen such that (3) holds for 1, ... , j - 1 then (18) implies (3) for 1, 2, ... , j provided 1/8k and Mk are sufficiently large fork?: j. This would give the desired statement W({8},{M}) c V({E},{N}).

First observe that by rotational invariance it suffices to prove (3) for x = (0, ... , 0, xn) with Xn > 0. We have for each n-tuple a,

(D0 u)(x) = (2n)-n { u(()(i()aei(x,<> d(. lr,.

H lxl > j so Xn > j we have for ( E r m

(23) e-(x,Im(} :::; (1 + ~~1)-jm, ekllm(l :::; (1 + ~~l)km

so if lal :::; Ni and if u satisfies (18) we have for constant C

(24) I(D0 u)(x)i :::; C f 8k 1 (1 + [1~1 2 + m2 (log(1 + 1~1))2 ) l/2)N;-Mk k=l R."

· (I+ IW(k-j)m(l + m) d~.

The terms with k < j decrease exponentially as m -+ +oo. We fix m such that their sum is < Ej/2. H Mk and 1/8k are large enough fork?: j then the sum of the terms with k ?: j is also :::; f.j /2. This would prove (3) for j. Also if Mj, Mj+l, ... and l/8i, l/8i+l> ... are increased from the previous step, (3) would remain valid for 1, 2, ... , j -1. This completes the induction and the proof of Theorem 3.4.

156 Chapter V. Fourier 'Iransforms and Distributions. A Rapid Course

§4 Differential Operators with Constant Coefficients

The description of the topology of 1J in terms of the range i5 given in Theorem 3.4 has important consequences for solvability of differential equations on Rn with constant coefficients.

Theorem 4.1. Let D :j; 0 be a differential operator on JR'l with constant coefficients. Then the mapping f -+ 1J f is a homeomorphism of 1J onto DTJ.

Proof. It is clear from Theorem 3.3 that the mapping f -+ D f is injective on V. The continuity is also obvious.

For the continuity of the inverse we need the following simple lemma.

Lemma 4.2. Let P :j; 0 be a polynomial of degree m, F an entire function on en and G = PF. Then

IF(()I ~ c sup IG(z +()I, ( E en' lzl:51

where C is a constant.

Proof. Suppose first n = 1 and that P(z) = 2::; akzk(am :j; 0). Let Q(z) = zm 2::; akz-k. Then, by the maximum principle,

(25) lamF(O)I = IQ(O)F(O) ~max IQ(z)F(z)i =max IP(z)F(z)l. jzj=1 lzl=1

For general n let A be ann x n complex matrix, mapping the balll(l < 1 in en into itself and such that

m-1 P(A() = a(f + L PA:((2, ... , (n)(f, a:/: 0.

0

Let

F1(() = F(A(), G1(() = G(A(), P1(() = P(A().

Then

and the polynomial

z-+ P1((1 +z, . .. ,(n)

has leading coefficient a. Thus by (1)

laF1 (()I ~max IG1 ((1 + z, (2, ... , (n)l ~ max IG1 (( + z)l. jzj=1 zEC"

lzl:51

§4 Differential Operators with Constant Coefficients 157

Hence by the choice of A

laF(()I $ sup IG(( + z)j zEC" izi::;t

proving the lemma.

For Theorem 4.1 it remains to prove that if Vis a convex neighborhood of 0 in V then there exists a convex neighborhood W of 0 in V such that

(26) /EV,D/EW=?jEV.

We take V as the neighborhood W ( { 8}, { M}). We shall show that if W = W( {€}, {M}) (same {M}) then (26) holds provided the Ej in {€} are small enough. We write u = Df sou(() = P(()f(() where Pis a polynomial. By Lemma 4.2

(27)

But lzl $ 1 implies

11{()1 $ C sup lu(( + z)l. lzi::;I

(1 + lz + (1)-M; $2M; (1 + j(l)-M;, lim (z +()I $ lim (I + 1,

so if C2M; ei Ej $ 8i then (26) holds. Q.e.d.

Corollary 4.3. Let D f. 0 be a differential operator on JRn with constant (complex) coefficients. Then

(28) DV'=V'.

In particular, there exists a distribution T on JRn such that

(29) DT=8.

Definition. A distribution T satisfying (29) is called a fundamental solution for D.

To verify (28) let L E V' and consider the functional D*u ~ L(u) on D*V (* denoting adjoint). Because of Theorem 3.3 this functional is well defined and by Theorem 4.1 it is continuous. By the Hahn-Banach theorem it extends to a distributionS E V'. Thus S(D*u) = Lu soDS = L, as claimed.

Corollary 4.4. Given f E V there exists a smooth function u on JRn such that

Du =f.


In fact, if T is a fundamental solution one can put u = f * T. We conclude this section with the mean value theorem of Asgeirsson

which entered into the range characterization of the X-ray transform in Chapter I.

Theorem 4.5. Let u be a C 2 function on BR x BR C !Rn x !Rn satisfying

{30)

Then

(31) f u(O,y)dw(y)= f u(x,O)dw(x) r<R. }IYI=r }lzl=r

Conversely, if u is of class C 2 near (0, 0) C !Rn x !Rn and if {91) holds for all r sufficiently small then

{32) (Lxu)(O, 0) = (Lyu)(O, 0).

Remark 4.6. Integrating Taylor's formula it is easy to see that on the space of analytic functions the mean value operator Mr (Ch. I, §2) is a power series in the Laplacian L. (See (44) below for the explicit expansion.) Thus (30) implies {31) for analytic functions.

For u of class C2 we give another proof. We consider the mean value operator on each factor in the product

JRn X !Rn and put

U(r, s) = (M[ M2u)(x, y)

where the subscript indicates first and second variable, respectively. If u satisfies {30) then we see from the Darboux equation {Ch. I, Lemma 3.2) that

a2 U n- 1 au a2U n- 1 au -+----=-+----. 8r2 r ar as2 s as

Putting F(r,s) = U(r,s)- U(s,r) we have

(33) n - 1 aF a2 F n - 1 aF _ O -r- ar - as2 - -8- as - '

(34) F(r,s) = -F(s,r).

After multiplication of (33) by rn-1 ~~ and some manipulation we get

-rn-1 Jl.. [(ap)2 + (ap)2] + 2Jl.. (rn-1 aF aF) _ 2rn-1 n-1 (ap)2 = 0 . as 8r 8s ar 8r 8s s as

§4 Differential Operators with Constant Coefficients 159

Consider the line M N with equation r + s = const. in the (r, s)-plane and integrate the last expression over the triangle OMN (see Fig. V.2).

Using the divergence theorem (Ch. I, (26)) we then obtain, if n denotes the outgoing unit normal, de the element of arc length, and · the inner product,

s

FIGURE V.2.

(35) { ( 2rn-1aFaF ,-rn-l [(aF) 2 + (aF) 2]) ·nde

loMN ar as ar as

2 !1 n-1n-1 (aF) 2 d d = r -- -- r s. OMN S as

On OM : n = (2-112 , -2-112), F(r, r) = 0 so ~~ + ~~ = 0.

On MN: n = (2-112,2-112).

Taking this into account, (35) becomes

2-t f rn-1 (aF - aF) 2 de+ 2 f { rn-1 n- 1 (aF) 2 dr ds = 0. }MN ar as jj OMN S as

This implies F constant so by (34) F = 0. In particular, U(r, 0) = U(O, r) which is the desired relation (31).

For the converse we observe that the mean value (Mr !)(0) satisfies (by

Taylor's formula)

where Cn ::/:- 0 is a constant. Thus

dMrf(O) r-1 dr -t 2cn(Lf)(O) as r -t 0.

Thus (31) implies (32) as claimed.

160 Chapter V. Fourier 'Transforms and Distributions. A Rapid Course

§5 Riesz Potentials

We shall now study some examples of distributions in detail. If a E C satisfies Re a > -1 the functional

{36)

is a well-defined tempered distribution. The mapping a -+ x+ from the half-plane Rea > -1 to the space S' (JR.) of tempered distributions is holomorphic (that is a-+ x+(<p) is holomorphic for each <p E S(JR.)). Writing

x+(<p) = 11 xa(<p(x)- <p(O)) dx + <p(0)1 + 100 xa<p(x) dx o a+ 1

the function a -+ x+ is continued to a holomorphic function in the region Rea> -2,a -#-1. In fact

<p(x)- <p(O) = x 100 <p'(tx) dt,

so the first integral above converges for Rea > -2. More generally, a -+ x+ can be extended to a holomorphic S' (JR.)-valued mapping in the region

Rea>-n-1, a"#-1,-2, ... ,-n,

by means of the formula

11 [ n-1 ] (37) x+(<p) =

0 xa <p(x)- <p(O)- x<p1(0) -· · ·- (: -l)!<p(n-1)(0) dx

100 n <p(k-1) (0) + 1 xa<p(x) dx + L (k- 1)!(a + k).

k=l

In this manner a -+ x+ is a meromorphic distribution-valued function on C, with simple poles at a = -1, -2, .... We note that the residue at a= -k is given by

(38) ( -1)k-1

Res xa = lim (a+ k)xa = t)(k-1). a=-k + a-+-k + (k- 1)!

Here J(h) is the hth derivative of the delta distribution 8. We note that x+ is always a tempered distribution.

Next we consider for Rea > -n the distribution ra on JR.n given by

§5 Riesz Potentials 161

Lemma 5.1. The mapping a: -+ ra extends uniquely to a meromorphic mapping from C to the space S' (JR.n) of tempered distributions. The poles are the points

a:= -n- 2h (hE z+)

and they are all simple.

Proof. We have for Rea: > -n

(39) ra(cp) =On 100 {Mtcp)(O)ta+n-l dt.

Next we note (say from (15) in §2) that the mean value function t -+ (Mtcp)(O) extends to an even C00 function on lR, and its odd order derivatives at the origin vanish. Each even order derivative is nonzero if cp is suitably chosen. Since by (39)

{40)

the first statement of the lemma follows. The possible (simple) poles of ra are by the remarks about xf. given by a:+ n- 1 = -1, -2, .... However if a:+n -1 = -2, -4, ... , formula {38) shows, since (Mtcp(O)){h) = 0, (hodd) that ra{cp) is holomorphic at the points a= -n- 1, -n- 3, ....

The remark about the even derivatives of Mtcp shows on the other hand, that the points a:= -n-2h (hE z+) are genuine poles. We note also from (38) and ( 40) that

(41) Res ra = lim (a:+ n)ra =On<>. a=-n a-+-n

We recall now that the Fourier transform T -+ T of a tempered distribution T on JR.n is defined by

T(cp) = T(lj))

We shall now calculate the Fourier transforms of these tempered distributions ra.

Lemma 5.2. We have the following identity

(42) ( a)~= 2n+a ~ r{(n + a:)/2) -or-n r 7r r( -a:/2) r , -a:- n (j. 2z+.

For a: = 2h (h E z+) the singularity on the right is removable and (42) takes the form

(43)


Proof. We use the fact that if 1j;(x) = e-lxl 212 then ;f(u) = (2·n")'~ e-lui2 /2

so by the formula J fg = J Jg we obtain for cp E S(llln ), t > 0,

I cp(x)e-tlxl2/2dx = (2tr)nl2rn/2 I cp(u)e-lul2/2tdu.

We multiply this equation by rl-a/2 and integrate with respect tot. On the left we obtain the expression

r( -a/2)T~ I cp(x)lxla dx'

using the formula

1oo e-tlxl2/2rl-a/2 dt = r(-~)T~Ixla'

which follows from the definition

r(x) = 100 e-ttx-l dt.

On the right we similarly obtain

(2tr)jr((n+a)/2)2~ I cp(u)lul-a-ndu.

The interchange of the integrations is valid for a in the strip -n < Rea < 0 so ( 42) is proved for these a. For the remaining ones it follows by analytic continuation. Finally, (43) is immediate from the definitions and (6).

By the analytic continuation, the right hand sides of (42) and (43) agree for a = 2h. Since

Resr(-a/2) = -2(-l)h/h! a=2h

and since by (40) and (38),

!~~ r-•-•( <p) = -ll. (2~)! [ (:. f (M'<p) L we deduce the relation

[( d ) 2h l r(n/2) (2h)! h dt (Mt(p) t=o = r(h + n/2) 22hh! (L cp)(O).

This gives the expansion

(44) Mt = ~ r(n/2) (tj2) 2h Lh

t'o r(h + n/2) h!

§5 lliesz Potentials 163

on the space of analytic functions so Mt is a modified Bessel function of tL112 • This formula can also be proved by integration of Taylor's formula (cf. end of §4).

Lemma 5.3. The action of the Laplacian is given by

(45) Lra = a(a+n-2)ra-2 , (-a-n+2 ~ 2z+) {46) Lr2-n = (2- n)On8 (n i: 2).

For n = 2 this 'Poisson equation' is replaced by

(47) L(logr) = 27!"8.

Proof. For Rea sufficiently large (45) is obvious by computation. For the remaining ones it follows by analytic continuation. For ( 46) we use the Fourier transform and the fact that for a tempered distribution S,

Hence, by {42),

n n 2 7!"2 27!"2 -

(-Lr -n)~ = 4r(~ _ 1) = r(~) (n- 2)8.

Finally, we prove (47). If t.p E V(JR.2 ) we have, putting F(r) = (Mrt.p)(O),

(L(logr))(t.p) = f logr(Lt.p)(x) dx = 100 {logr)27rr(Mr Lt.p)(O) dr. }R2 o

Using Lemma 3.2 in Chapter I this becomes

100 log r 27rr(F" (r) + r-1 F' (r)) dr,

which by integration by parts reduces to

[ logr(27rr)F'(r)J:- 271" 100 F'(r) dr = 27rF(O).

This proves ( 4 7).

Another method is to write (45) in the form L(a-1 (ra -1)) = ara-2 •

Then (47) follows from (41) by letting a-+ 0. We shall now define fractional powers of L, motivated by the formula

so that formally we should like to have a relation

(48)


Since the Fourier transform of a convolution is the product of the Fourier transforms, formula (42) (for 2p =-a-n) suggests defining

(49)

where rr is the Riesz potential

(50) (J'Y f)(x) = Hnl(T) Ln J(y)jx- yj-y-n dy

with

(51) H ( ) 2"~ n r(t) n 'Y = 11"2 r(T).

Note that if-"( E 2z+ the poles of r(Tj2) cancel against the poles of r-y-n because of Lemma 5.1. Thus if 'Y- n fl. 2.z+ we can write

By (12) and Lemma 5.2 we then have

(53)

as tempered distributions. Thus we have the following result.

Lemma 5.4. Iff E S(IRn) then 'Y ~ (P f)(x) extends to a holomorphic function in the set Cn = {'Y E C: "(- n fl. 2Z+}. Also

(54)

(55)

limP/=/, -y-+0

P Lf = LP f = -P-2 f.

We now prove an important property of the Riesz' potentials. Here it should be observed that rr f is defined for all f for which (50) is absolutely convergent and 'Y E Cn.

Proposition 5.5. The following identity holds:

Io:(I13 f)= Jo:+/3 f for f E S(IRn), Rea, Re,B > 0, Re(a + ,8) < n,

Io:(Jt3 f) being well defined. The relation is also valid if

f(x) = O(lxi-P) for some p >Rea+ Re,B.

Proof. We have

Io:(I13 f)(x) = Hn~a) I lx- zjo:-n (Hnl(,B) I f(y)jz- yjt3-n dy) dz

= Hn(a)~n(,B) I f(y) (I lx- zio:-nlz- Yi/3-n dz) dy.


The substitution v = (x- z)/!x- yJ reduces the inner integral to the form

(56) Jx- yJa+.B-n { Jv!a-n!w- vJ.B-n dv' }JR ...

where w is the unit vector (x- y)/lx- yJ. Using a rotation around the origin we see that the integral in (56) equals the number

(57)

where e1 = (1, 0, ... , 0). The assumptions made on a and (3 insure that this integral converges. By the Fubini theorem the exchange order of integrations above is permissible and

(58)

It remains to calculate cn(a,(3). For this we use the following lemma which was already used in Chapter I, §2. As there, let S* (JRn) denote the set of functions in S(JRn) which are orthogonal to all polynomials.

Lemma 5.6. Each Ia(a E Cn) leaves the space S*(JRn) invariant.

Proof. We recall that (53) holds in the sense of tempered distributions_:. Suppose now f E S* (JRn). We consider the sum in the Taylor formula for f in Jul ~ 1 up to order m with m > JaJ. Since each derivative of J vanishes at u = 0 this sum consists of terms

1!31 =m

where Ju*l ~ 1. Since Ju.BJ ~ JuJm this shows that

(59)

Iterating this argument with 8i(luJ-aJ(u)) etc. we conclude that the limit relation (59) holds for each derivative D.B(iul-a J(u)). Because of (59), relation (53) can be written

(60) f (I"' f)~(u)g(u) du = f Jul-a f(u)g(u) du, g E S, }JRn }JRn

so (53) holds as an identity for functions FE S*(JRn). The remark about D.B(iui-a J(u)) thus implies (Ia f)~ E So so I"' f E S* as claimed.

We can now finish the proof of Prop. 5.5. Taking fa E S* we can put f = [.13 fa in (53) and then

(I"'(J.B fo))~(u) = (J.B fo)"'(u)Jui-a = fo(u)Jui-a-.13

= (I':Jt+.BJo)"'(u).


This shows that the scalar factor in (58) equals 1 so Prop. 5.5 is proved. In the process we have obtained the evaluation

{ lvla-nle1 - vj.B-n dv = Hn(a)Hn(f3) . JR,. Hn(a + (3)

We now prove a variation of Prop. 5.5 needed in the theory of the Radon transform.

Proposition 5.7. Let 0 < k < n. Then

rk(Jkf) = f

if f(x) = O(lxj-N) for some N > n.

Proof. By Prop. 5.5 we have if f(y) = O(IYI-N)

(61) Ia ( Jk f) = J<k+k f for 0 < Rea < n - k .

We shall prove that the function cp = Jk f satisfies

(62) sup jcp(x)jjxjn-k < oo. X

For an N > n we have an estimate lf(y)l :::; eN(1 + !yi)-N where eN is a constant. We then have

( f f(y)lx- Yik-n dy) :::; eN f 1 (1 + IYD-Nix- Ylk-n dy }Rn Jlx-yl$2lxl

+eN { 1 (1 + jyi)-Njx- Ylk-n dy · J1x-yi2':2ixl

In the second integral, jx- ylk-n :::; ( ~ )k-n so since N > n this second

integral satisfies (62). In the first integral we have IYI ~ ~ so the integral is bounded by

(1 + ~) -N { jx- yjk-n dy = (1 + ~) -N { !zlk-n dz 2 Jlx-yl$~ 2 J!zl$~

which is O(lxi-Nixlk). Thus (62) holds also for this first integral. This proves (62) provided

f(x) = O(lxi-N) for some N > n.

Next we observe that I a ( cp) = Ja+k (f) is holomorphic for 0 <Rea< n- k. For this note that by (39)

(Ja+k !)(0) = 1 { J(y)jyja+k-n dy Hn(a + k) }R,.

= 1 On ['X) (Mt f)(O)ta+k-l dt. Hn(a + k) Jo


Since the integrand is bounded by a constant multiple of t-N ta+k-l, and

since the factor in front of the integral is harmless for 0 < k + Rea < n, the holomorphy statement follows.

We claim now that Ia(cp)(x), which as we saw is holomorphic for 0 < Rea < n - k, extends to a holomorphic function in the half-plane Rea < n- k. It suffices to prove this for x = 0. We decompose cp = cp1 + cp2 where cp1 is a smooth function identically 0 in a neighborhood lxl < t: of 0, and cp2 E S(Rn ). Since cp1 satisfies (62) we have for Rea < n- k,

If 'Pl(x)lxiRea-ndxl < C [oo lxlk-nlxiRea-nlxln-ldlxl

= C [oo lxiRea+k-n-ldlxl < oo

so Iacp1 is holomorphic in this half-plane. On the other hand Iacp2 is holomorphic for a E Cn which contains this half-plane. Now we can put a = -k in (60) and the proposition is proved.

Denoting by C N the class of continuous functions f on Rn satisfying f(x) = O(lxi-N) we proved in (62) that if N > n, 0 < k < n, then

(63)

More generally, we have the following result.

Proposition 5.8. If N > 0 and 0 < Re7 < N, then

where s = min(n, N)- Re7 (n =j:. N).

Proof. Modifying the proof of Prop. 5. 7 we divide the integral

into integrals I1, h and Is over the disjoint sets

A1 = {y: IY- xi$ ~lxl}, A2 = {y: IYI < ~lxl,

and the complement As= an- A1- A2. On A1 we have IYI;::: !lxl so

so

(64)


On Az we have lxl + !lxl ~ !x- Yl ~ !lx! so

!x- Y!Re-y-n $ C!x!Re-r-n, C =canst ..

Thus

H N > n then

H N < n then

In either case

(65)

On A3 we have (1 + !y!)-N $ !YI-N· The substitution y = lx!u gives (with e = xflx!)

(66) 13 $ !x!Re-y-N r 1 1 !ui-Nie- u!Re-y-n du = O(lx!Re-y-N). Jlul2:2,1e-u12:2

Combining (64)-(66) we get the result.


§1-2 contain an exposition of the basics of distribution theory following Schwartz [1966]. The range theorems (3.1-3.3) are also from there but we have used the proofs from Hormander [1963]. Theorem 3.4 describing the topology of 1J in terms of i5 is from Hormander [1983], Vol. II, Ch. XV. In the proof we specialize Hormander's convex set K to a ball; it simplifies the proof a bit and requires Cauchy's theorem only in a single variable. The consequence, Theorem 4.1, and its proof were shown to me by Hormander in 1972. The theorem appears in Ehrenpreis [1956].

Theorem 4.5, with the proof in the text, is from Asgeirsson [1937]. Another proof, with a refinement in odd dimension, is given in Hormander [1983], Vol. I. A generalization to Riemannian homogeneous spaces is given by the author in [1959]. The theorem is used in the theory of the X-ray transform in Chapter I.

§5 contains an elementary treatment of the results about Riesz potentials used in the book. The examples xt are discussed in detail in Gelfand-Shilov [1959]. The


potentials 1>. appear there and in Riesz [1949] and Schwartz [1966]. In the proof of Proposition 5. 7 we have used a suggestion by R. Seeley and the refinement in Proposition 5.8 was shown to me by Schlichtkrull. A thorough study of the composition formula (Prop. 5.5) was carried out by Ortner [1980] and a treatment of Riesz potentials on £P-spaces (Hardy-Littlewood-Sobolev inequality) is given in Hormander (1983], Vol. I, §4.

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Notational Conventions 185

Notational Conventions

Algebra As usual, lR and C denote the fields of real and complex numbers, respectively, and .Z the ring of integers. Let

If a E C, Rea denotes the real part of a, Im a its imaginary part, jaj its modulus.

If G is a group, A C G a subset and g E G an element, we put

Ag = {gag-1 : a E A} ,gA = {aga-1 :a E A}.

The group of real matrices leaving invariant the quadratic form

is denoted by O(p, q). We put O(n) = O(o, n) = O(n, o), and write U (n) for the group of n x n unitary matrices. The group of isometries of Euclidean n-space JR2 is denoted by M ( n).

Geometry The (n -I)-dimensional unit sphere in JRn is denoted by sn-1 ,

fln denotes its area. The n-dimensional manifold of hyperplanes in JRn is denoted by pn. If 0 < d < n the manifold of d-dimensional planes in JRn is denoted by G(d, n); we put Gd,n ={a E G(d, n) : o E a}. In a metric space, Br(x) denotes the open ball with center x and radius r; Sr(x) denotes the corresponding sphere. For pn we use the notation t3A(0) for the set of hyperplanes~ C llr" of distance< A from 0, a A for the set of hyperplanes of distance = A. The hyperbolic n-space is denoted by Hn and the n-sphere by sn.

Analysis If X is a topological space, C(X) (resp. Cc(X)) denotes the sphere of complex-valued continuous functions (resp. of compact support). If X is a manifold, we denote:

cm(X) = {complex-valued m-times continuously} differentiable functions on X

C00 (X) = £(X) = nm>oCm(X). C;;'"(X) = D(X) = Cc(X) n c=cx). D' (X) = {distributions on X} .

£'(X) = {distributions on X of compact support} .

DA(X) = {! E D(X) : support f C A}. S(JRn) = {rapidly decreasing functions on JRn } .

S'(JRn) = {tempered distributions on JRn} .

The subspaces DH, SH, S*, S0 of S are defined on pages 5 and 9.

186 Notational Conventions

While the functions considered are usually assumed to be complexvalued, we occasionally use the notation above for spaces of realvalued functions.

The Radon transform and its ~ual are denoted by f -+ [, cp -+ cp, the Fourier transform by f -+ f and the Hilbert transform by 1-l.

F\ I~, I; and It denote Riesz potentials and their generalizations. Mr the mean value operator and orbital integral, L the Laplacian on JR.n and the Laplace-Beltrami operator on a pseudo-Riemannian manifold. The operators 0 and A operate on certain function spaces on pn; 0 is also used for the Laplace-Beltrami operator on a Lorentzian manifold, and A is also used for other differential operators.

INDEX

Abel's integral equation, 11 Adjoint space, 121 Antipodal manifold, 111 Approximate reconstruction, 48 Asgeirsson's mean-value theorem,

39, 158

Cauchy principal value, 19 Cauchy problem, 42 Cayley plane, i14 Cone

backward, 43 forward, 43 light, 45, 128, 131 null, 128 retrograde, 130, 131 solid, 45

Conjugacy class, 62 Conjugate point, 111 Curvature, 124, 126 Cusp forms, 79

Darboux equation, 16, 91, 137 Delta distribution, 27, 160 Dirichlet problem, 71 Distribution, 149

convolution, 150 derivative, 149 Fourier transform of, 151 Radon transform of, 23 support of, 149 tempered, 149

Divergence theorem, 13 Double fibration, 55 Duality, 55 Dual transform, 2, 74

Elliptic space, 93 Exponentially decreasing functions,

120

Fourier transform, 150 Fundamental solution, 157 Funk transform, 63, 64

Generalized sphere, 62 Gram determinant, 127 Grassmann manifold, 29, 74

Harmonic line function, 40 Hilbert transform, 18 Horocycle, 67 Huygens' principle, 45, 144, 145 Hyperbolic space, 85, 96

Cayley, 118 complex, 118 quaternian, 118 real, 118

Incident, 53 Inductive limit, 147 Invariant differential operators, 3 Inversion formula, 15, 25, 27, 29,

41,64,66-68,91,94,96, 97, 99, 101, 103

Isometry, 3, 21, 123 Isotropic, 123, 124

geodesic, 129 space, 120 vector, 128

John's mean value theorem, 40, 72

Laplace-Beltrami operator, 123 Laplacian, 3 Light cone, 45, 128, 131 Lorentzian, 123

manifold, 123 structure, 123

188 Index

Mean value operator, 8, 91, 112, 133

Mean value theorem, 39, 40, 158 Modular group, 79 Multiplicity, 112

Null cone, 128

Orbital integrals, 62, 128, 133

Paley-Wiener theorem, 14, 152 Plancherel formula, 20, 151 Plane wave, 1

normal of, 41 Plucker coordinates, 38 Poisson

equation, 163 integral, 71 kernel, 71

Projective spaces Cayley, 114 complex, 114 quaternian, 114 real, 114

Property ( S), 49 Pseudo-Riemannian

manifold, 123 structure, 123

Radial function, 16 Radiograph, 47 Radon transform, 2, 58

d-dimensional, 28 for a double fibration, 57 of distributions, 60 of measures, 60

Rapidly decreasing functions, 4, 119, 120

Residue, 160 Retrograde cone, 130

solid, 130, 131 Riesz potential, 160 Riesz potentials

generalized, 137

Seminorms, 4, 147 Source, 43 Spacelike, 128

geodesic, 129 vectors, 128

Spherical function, 99 Spherical slice transform, 108 Spherical transform, 99, 100 Support theorem, 2, 10, 85

Theta series, 79 Timelike, 128

geodesic, 129 vectors, 128

Totally geodesic, 83, 111 lransversality, 55 Two-point homogeneous, 83

Wave, 1, 43 incoming, 43 outgoing, 43

Wave equation, 42 Wave operator, 134

X-ray, 46 reconstruction, 46 transform, 28, 37, 66, 99, 119

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137 VIGNERAS. Representations I-modulaires d'un groupe reductif p-adique avec I 'I= p

138 BERNDT/DIAMOND/HILDEBRAND ( eds.) Analytic Number Theory. Vol. 1 In Honor of Heini Halberstam

139 BERNDT/DIAMOND/HILDEBRAND (eds.) Analytic Number Theory, Vol. 2 In Honor of Heini Halberstam

140 KNAPP. Lie Groups Beyond an Introduction

141 CABANEs. Finite Reductive Groups: Related Structures and Representations

142 MoNK. Cardinal Invariants on Boolean Algebras

143 GONZALEZ-VEGA/RECIO. Algorithms in Algebraic Geometry and Applications

144 BELLAICHFJRISLER (eds). Sub-Riemannian Geometry

145 ALBERT/BROUZET/DUFOUR (eds.) Integrable Systems and Foliations. Feuilletages et Systemes Integrables.

146 JARDINE. Generalized Etale Cohomology 147 DIBIASE. Fatou TypeTheorems. Maximal

Functions and Approach Regions 148 HUANG. Two-Dimensional Conformal

Geometry and Vertex Operator Algebras 149 So~u. Structure of Dynamical Systems.

A Symplectic View of Physics 150 SHIOTA. Geometry of Subanalytic and

Semialgebraic Sets 151 HUMMEL. Gromov's Compactness

Theorem for Pseudo-holomorphic Curves 152 GROMOV. Metric Structures for

Riemannian and Non-Riemannian Spaces 153 BUEScu. Exotic Attractors: From

Liapunov Stability to Riddled Basins 154 BOTICHER/KARLOVICH. Carleson Curves,

Muckenhoupt Weights, and Toeplitz Operators

155 DRAGOMIR/ORNEA. Locally Conformal Kahler Geometry

156 GUIVARCH/JI/TAYLOR. Compactifications of Symmetric Spaces

157 MURTY/MURTY. Non-vanishing of Lfunctions and Applications

158 TIRAONOGAN/WOLF(eds). Geometry and Representation Theory of Real and p-adic Groups

159 THANGA VELU. Harmonic Analysis on the Heisenberg Group

160 KAsHIWARA/MATSUO/SAITO/SATAKE (eds). Topological Field Theory, Primitive Forms and Related Topics

161 SAGAN/STANLEY (eds). Mathematical Essays in Honor of Gian-Carlo Rota

162 ARNOLD/GREUEI}STEENBRINK. Singularities. The Brieskom Anniversary Volume

163 BERNDT/SCHMIDT. Elements of the Representaiton Theory of the Jacobi Group

164 RousSARIE. Bifurcations of Planar Vector Fields and Hilbert's Sixteenth Problem

165 MIGUORE. Introduction to Liaison Theory and Deficiency Modules

166 EuAs/GIRAI}MIRO-ROIG/ZARZUELA ( eds ). Six Lectures on Commutative Algebra

167 FACCHINI. Module Theory 168 BALOG/KATONA/SZA'SZ/RECSKI (eds).

European Congress of Mathematics, Budapest, July 22-26, 1996. Vol. I

169 BALoo/KATONA/SZA'SZ/RECSKI (eds). European Congress of Mathematics, Budapest, July 22-26, 1996. Vol. II

168/169 Sets Vols I, II 170 PATERSON. Groupoids, Inverse

Semigroups, and their Operator Algebras

171 REzNIKOV/SCHAPPACHER (eds). Regulators in Analysis, Geometry, and Number Theory

172 BRYUNSKI/BRYUNSKI/NISTOR/ TSYGAN/ Xu ( eds ). Advances in Geometry

173 DRAXLER/MICHLER/RINGEL ( eds.) Computational Methods for Representations of groups and algebras; Euroconference in Essen

174 GoERSS/JARDINE. Simplicial Homotopy Theory

175 BANUELOS/MOORE. Probabilistic Behavior of Harmonic Functions

176 BASS/LUBOTZKY. Tree Lattices 177 BIRKENHAKE/LANGE. Complex Tori 178 PUIG. On the Local Structure of

Morita and Richard Equivalence Between Brauer Blocks

179 RuiZ/JosE. Differential Galois Theory and Non-integrability of Hamiltonian Systems

The Radon Transform

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