Estimates for spherical harmonics
Citation for published version (APA):van Berkel, C. A. M., & Eijndhoven, van, S. J. L. (1989). Estimates for spherical harmonics. (RANA : reports onapplied and numerical analysis; Vol. 8920). Eindhoven: Technische Universiteit Eindhoven.
Document status and date:Published: 01/01/1989
Document Version:Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)
Please check the document version of this publication:
• A submitted manuscript is the version of the article upon submission and before peer-review. There can beimportant differences between the submitted version and the official published version of record. Peopleinterested in the research are advised to contact the author for the final version of the publication, or visit theDOI to the publisher's website.• The final author version and the galley proof are versions of the publication after peer review.• The final published version features the final layout of the paper including the volume, issue and pagenumbers.Link to publication
General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.
• Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal.
If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, pleasefollow below link for the End User Agreement:
www.tue.nl/taverne
Take down policyIf you believe that this document breaches copyright please contact us at:
providing details and we will investigate your claim.
Download date: 27. Feb. 2020
Eindhoven University of Technology Department of Mathematics and Computing Science
RANA 89-20
September 1989
ESTIMATES FOR
SPHERICAL HARMONICS
by C.A.M. van Berkel
and
S.J.L. van Eijndhoven
Repons on Applied and Numerical Analysis
Department of Mathematics and Computing Science
Eindhoven University of Technology
P.O. Box 513
5600 MB Eindhoven
The Netherlands
Summary
ESTIMATES FOR SPHERICAL HARMONICS by
C.A.M. van Berkel and
S.J.L. van Eijndhoven
In this paper we present Loo- and Lrestimates for spherical harmonics in which partial differentiation operators and multiplication operators are involved. Our results are compared with classical results of Stein, Seeley and Calderon and Zygmund.
A.M.S. Classifications: 26DlO, 33A45. Key Words: spherical harmonics, homogeneous polynomials, reproducing kernels.
-2-
1. Introduction
The present discussion starts with the space IRq. q ~ 3, provided with the Euclidean inner
product X· Y = Xl Y 1 + ... + Xq Yq. By cfl-l we denote the surface measure on the unit sphere Sq-l in IRq, normalized by the relation
dx = r q - l dr dcfl- l (co) (i)
where dx = dxl ... dxq • So we have
coq := J dcfl-1 = 21tq12 r(q 12). S,-1
(ii)
Consider the Hilbert space LZ(Sq-l) of square integrable functions on Sq-I with the
natural inner product
(f.g>L2(Sf-I ):= J f(~) g(~) dcfl-l(~) ,f.g E LZ(Sq-I).
5q- 1
We introduce the following spaces of polynomials in q variables
pq : the linear space of all polynomials
(iii)
HZt : the linear space of all m-homogeneous hannonic polynomials (iv)
yZt : the linear space of all restrictions of polynomials in HZt to Sq-l .
All spaces can be endowed with the Lz(sq-l)-inner product. The vector spaces HZt and
yZt are finite dimensional each with dimension d1,. given by
dZt = (2m+q-2)(m+q-3)! . (q-2)! m!
And there is the orthogonal decomposition
... Lz(Sq-l):::: $ YZt·
m=O
In pq we introduce another inner product,
(PloPZ)pq :=Pl("V)P2(X) Ix=o, Pl,PZ E pq
where V represents the gradient vector [ a; 1 ••••• a:,],
(v)
(vi)
(vii)
For harmonic homogeneous polynomials this pq-inner product is related to the LZ(Sq-l)
inner product as follows. Let hI and h2 be harmonic homogeneous polynomials of degree
m and n respectively. Then
- 3 -
(viii)
(for a proof we refer to [Ma, Lemma 2.14 and Lemma 2.25] or [Be, Chapter 1 D· The Gegenbauer polynomials C~, m E iNo, A> -I, introduced in [MOS, p. 218 ff.] obey
the orthogonality relations
Now for mE iNo and q~ 3 we set
P'/n = m! (q - 3)! C7j2-1. (q+m-3)!
Then P'/n(l) = 1 and according to (ix)
I 00 f P'/n(t) P%(t) (l-t2 )t(q-3) dt = q ~lIm. -I ooq_1 d'/n
It is a well-known result, cf. [Mil, Lemma 7, p.14]. that for all I E Y'/n and S E Sq-l
dq
I(s) = ~ J P'/n(s' 00)/(00) doll-I (00). ooq sq-I
(ix)
(x)
(xi)
(xii)
dq
So (S. 00) f-7 ..-!'!... P'/n(s' 00) is the reproducing kernel of the Hilbert space Y'/n and for any ooq
orthonormal basis {e'ln,j : 1:::;; j:::;; d'ln} of Y'ln we have
(xiii)
Using the reproducing kernel property of the P'In we obtain for all h E H'In and; E Sq-l
[dq ]\i [dqj\i
I he;) I :::;; 00: P'/n(s- S) IIhIlL2(SQ-I) = ~ IIhIlL2(Sq-l) , (xiv)
In this paper we present various Loo- and Lz-estimates for spherical harmonics. We conclude the introduction with some notations concerning multi-indices and partial derivatives. By 10k we denote the multi-index with components eJ =~kj, k,j E {l, ... , q},
and 10 = (1, ... , 1). Furthermore, for all k E IN 0, X E IR q, and for all multi-indices
a, ~ E IN6
I al = al + ... + aq
a+k~= (al+k~), ... ,aq+k~q)
as: ~ :~ al s: ~1>"" aqs: ~q
(a)p = (al)!lJ· ... • (aq)p,
-4-
here (. ) j denotes the Pochhammer symbol, which is defined by
(b),'= r(b+j) . 012 J' r(b) ,J = , , , ....
Utilizing the inequalities
[ n+nm~kk-ll «::; [n +nm] _ _ ,n;:::k;:::O ,m:?:.I;:::O,
it follows that for all multi-indices a, ~ E !Nt, with ~;::: £,
(~)aS: (1~I-q+1)lal'
(xv)
(xvi)
(xvii)
(xviii)
By akfwe denote the partial derivative of fwith respect to the k-th variable. As usual, for
all multi-indices a E !Nt, we write
Clearlyaa maps H~ onto H~-Ial' where H% is taken to be the trivial space for n < O.
2. Loo- and Lrestimates for spherical harmonics
We start with integrating monomials oof'> over the unit sphere Sq-I.
Lemma 1. Let ~ E JN6 be a multi-index.
(i) If~jisoddforsomejE {l, ... ,q},then J oof'>da'l-l(oo)=O. sq-I
(ii) For all multi-indices a E !N'!, satisfying as: ~, we have
(a+1. £)p.-a J OOlP da'1-1(00) = z J 002a da'l-I (00).
S,-I (Ial q)Ip.-a1 S,-I
Proof
(xix)
Consider the integral J ooP da'l-l(oo). Substituting OO=tel + ",,1-t2 'fl, with -1 s: tS: 1 and sq-I
T\ E span {ez, ... , eq } we derive
-5
1 J ro?> d<fl-I(ro)= J t?>'(l_t2)t(q+IJ3H,-3) dt J T1~ ••. T1!q d<fl-2 (T1) = Sq-l -I Sq-2
{
o , if~1 is odd = r(l(Pl+l»r(l(q+I~I-~l-l»
2 ret (~+ I PI» /2 T1~ ... T1~q daq-2
(T1) , if ~1 is even.
Now induction arguments yield the first part of the lemma. FurthelIDore, with the aid of the above fOlIDula and the relation z rez) = rez + 1) we obtain
~1 +l J Ol2@+£') d<fl-1 (ol) = ; J ol2P d<fl-1 (ro). sql 1 PI + 2 q sql
And from this fOlIDula we deduce for all m E IN 0 and k E {I, ... , q},
(Pk+t )m J Ol2<J3+me') daq-1(Ol) = 1 J Ol2f} d<fl-1(ro). sq-l (IPI +2 q )m sq-l
U sing these integral relations the second part of the lemma follows by induction. I]
Taking a. = 0 in the preceding lemma, we get explicit expressions for integrals of monomials:
Corollary 2. For all multi-indices P E lNg,
27tq12 (l E)?> J Ol2?> d<fl-1 (ro) = 1 : . sq-l r(2 q) (2 q)I?>1
I]
Theorem 3.
Let p be an m-homogeneous polynomial and let a E IN6 be a multi-index. Then
(m+l)lal IIx o. pllt2(sQ-l):::; (}2 IIpllt2(sQ-l).
m+2 q)lal
Proof.
The proof is by induction to the length of a E INg. Obviously equality holds for a. = O.
Suppose the estimates are valid for all multi-indices a E INg with length I a 1 ::;; n for some n 2 O. Let P E INg be a multi-index with length 1131 = n + 1. Then there exists j E {l, ... , q} such that Pj 2 1. Since p is an m-homogeneous polynomial, x 2$-ei) Ip 12 is a 2(m +n)-homogeneous polynomiaL Therefore, x2(J3-ei) Ip 12 is a linear combination of monomialsxY with Iyl =2(m+n):
-6-
x 2(]Hi) Ip(x)1 2 = L c(Y)xY• Iyl =2(m+n)
With the aid of Lemma 1 we estimate
J ro2P lp(ro)1 2 dcfl-1(ro)= L C(y) J roy+-2r!dcfl-1(ro)= S9-1 Iyl =2(m+n) So-I
YI + 1 J == L c(y) roy dcfl-J (ro) S; Iyl =2(m+n) 2(m +n) + q S.-I
S; +1 J ro2(]Hi) Ip(ro)1 2 dcfl-I (ro). 2(m +n) + q S.-I
SO using the induction hypothesis we obtain
(m+i)n+l J ro2P Ip(ro)12 dcfl-1(ro)S; J Ip(ro)1 2 dcfl-1(ro) S,-1 (m +i q)n+l S.-I
which completes the proof.
Note that equality in the above theorem holds if (l = m Ek and p(x) = (Xk)m.
n
Next we pay attention to estimates for partial derivatives of spherical harmonics. But we gather some auxiliary results first.
Lemma 4. Let p be a homogeneous polynomial of degree m. Then for each k e {I, ... , q}
J (akP)(ro)dcfl-1(ro)==(m+q-l) J rokP(ro)dcfl-1(ro). S'I-I S.-I
Proof.
We may as well assume that k == 1. Let 'II denote a compactly supported differentiable function on /R+. Using the homogeneity of a1 p we have
J 'I'(r)rq- 1 dr J (a1P)(ro)dcfl-1(ro)= o S.-I
== f {J Ixl-m+I 'V(lxl)(a1P)(x)dxd dx2 ... dxq . /Rq-l IR
Applying partial integration the latter integral equals
J {f Ixl-m- 1 [(m-l)'II(lxl)-lxl ¥(lxl)]XIP(x)dxddx2 '" dxq = IRq-I IR
= J [(m-l)'V(r)-r¥(r)]rq-1dr J rolP(ro)dcfl-1(ro) o ~~
and the result follows.
- 7 -
Corollary S. Let f and g be homogeneous polynomials of degree m and n. Then for each k E {I, ... , q}
Proof.
Putting p = f· Ii in Lemma 4, we are done. o
Note that the above relation is a special case of the integral formula established in [Ma, Theorem 1.16]. However, our proof is based on simpler arguments.
Lemma 6.
Let hE Ht and let k E {I, ... , q}. Then we have
ndkhlll,(sq-I) = (2m +q - 2) [em ++ q) IIXk hlll;2(sQ-I) - + IIhlll;2(s4-1)].
Proof.
Let f = hand g = dA; h in the previous corollary. Then we derive, writing (. " ) instead of (. " k,(S4 1)
ndA;hIlL2(S4-1) (i3th, dkh)=(dA;h, dA;h)+(h, dth)=
= (2m+q-2) (xA;h, dA;h) =
= (2m +q - 2). + [(dA;(XA; h), h) + (Xk h, dkh) - (h,h)] =
= (2m +q -2) [em +t q) (xt h, h) - + (h,h)]
where in the last step Corollary 5 is used again.
As a consequence of Lemma 6 and Theorem 3 we get immediately
Corollary 7.
Leth E Ht and letk E {I, ... , q}. Then
ndkhIl1,(S4-'):S; m(2m +q - 2) Ilhlll,(S4-1).
(]
(]
We recall that for each multi-index a E IN'!" the operator da maps Ht onto Ht-Ial. Now we arrive at the main theorem of this paper covering Loo(Sq-l) - and L 2(Sq-l) - estimates for partial derivatives of spherical harmonics.
- 8 -
Theorem 8.
(i) Let 0. E INZ. Then for all h E HZt
lIaahlit SO-I :s;; m!(2m+q-2)!! IIhllt2(so-I).
2( ) (m-lo.l)! (2m+q-2-210.1)!!
(ii) Let 0. E INZ. Then for all h E HZt and ~ E Sq-l
[ dZt-lal m! (2m+q-2)!! ]Ih
l(aah)(~)I:S;; (i)q (m-lo.l)! (2m+q-2-210.1)!! IIhIlL2(SQ-I).
Proof.
The first part of the theorem follows from Corollary 7 and a simple induction argument. The second part is a direct consequence of the inequality (xiv) presented in the introduction and the first part of this theorem. 11
We now give another, very elegant proof of Theorem 8(i) wltt!rt! the correspondence (viii) between the pq -inner product and the L 2(Sq-l )-inner product is utilized.
Second proof of Theorem 8(i).
Let 0. E INZ with 10.1 :s;; m. Since hE HZt, h can be written as
hex) = L b(~) xj} 1j}I=m
and so
(aa h) (x) = L (~-o.+£)a b(~)xlMx. 1j}I=m
From relation (viii) it follows that
n-q/2 2m-
1 r(m +t q) (h,hk2(sO-I) = (h,h)pq = L ~! I b(~) 12. 1j}I=m
The polynomial aa h belongs to HZt-lal and so, applying (viii) once more and using (xviii) thereafter, we estimate
n-ql2 2m-lal-l r(m - 10.1 +t q) (aa h, aa hk2(so-l) =
= (aa h, aah)pq = L ~! (~-a+£)a Ib(~)12:s;; 1j}I=m
:s;; (m-Ial +l)lal L ~! Ib(~)12= 1j}I=m
m! -q12 2m- 1 r( 1) (h h) = ·n m+-q, L (SO-I) , (m - 10. I)! 2 2
yielding the wanted result. n
-9-
Using Stirling's fonnula we obtain from Theorem 8(ii) the following asymptotic fonnula.
Corollary 9. For each multi-index a E IN6 there exists a constant Cq•1al > 0 such that for all hE H;k and ~ E Sq-l
[]
Although the result of Corollary 9 is known (cf. [CZ, fonnula (4)] or [Se, Theorem 4b]) our proofs are more concise and based on elementary techniques.
3. Comparison with corresponding results.
Stein has proved the following pointwise estimate, cf. [St, p.276].
For any multi-index a E IN6 there exists a constant Cq,a such that for all hE H;k and for all ~ E Sq-l
(xx)
From Corollary 9 we see that instead of mtq+1al , in Stein's estimate, we can take
mtq-l+lal. This improvement by a factor m has consequences for the continuity and com
pactness of the differential operators aa on certain function spaces of hannonic functions (cf. [Li2, Theorem 7]). Liu and Martens proved in [LM, Theorem 4] the following result.
Let k E {l, ... , q}, let~ E Sq-l and let hE H;k. Then
I (d. h)(l;) I ~ m [ ~ ] " IlhIlL,~ .. ). (xxi)
We shall show that this estimate is best possible. According to the estimate of Theorem 8(ii), we get
I (ak h) (~) I ::; [ 2m +q -4] ~ . m [ d;k] 'h IIhIlL2(Sq-l).
m+q-3 ffiq (xxii)
So the estimate (xxi), settled by Liu and Martens, is about a factor 12 sharper than ours. In their proof they explore the Gegenbauer polynomials. Using an orthonormal basis {4,1 : 1::; j::; d'ln} in H'In they write
- 10-
~ h(~) = L (h, e7n,jk1(SQ-l) e7n,j(~).
j=l
Hence, by the Cauchy-Schwarz inequality,
[
tf! ]Ih 1 Cdlc h) (~) 1 ~ IIhIlL2(S9 I) ~ 1 (die e'ln,J) (~) 12
J=1
It turns out that
from which the result can be obtained.
Take k = I and ~ = el = (1,0, ... ,0). We define ho E H7n by
Then
~----ho = L (a l e7n,j) (el) e7n,j'
j=l
I (a, ho) (e,)1 =m [ roo] "llhOIIL,(s'-»'
(xxiii)
(xxiv)
(xxv)
(xxvi)
(xxvii)
So their estimate is best possible. This answers a question raised in LLM]. We note that in [LM] no estimates for I (d<X h) (~) I with I al > 1 are given. In fact, their techniques are not appropriate for deriving these estimates. Finally we present another method, based on the reproducing kernel of Y7n-l, to obtain supremum norm estimates for the partial derivatives dlch. Let h E H7n and let k E {I, ... , q}. Using the reproducing kernel of Y7n-l, the orthogonality relations of spherical harmonics, Corollary 5 and the Cauchy-Schwarz inequality we get for all ~ E Sq-l,
(xxviii)
If we take ~ = elc and substitute ill = telc + ;}1-t2 11 in the integral, a tedious calculation of the integral yields
[ dq ]14
I (dlc h) (ek) I ~ B7n 00: IIhIl L2(S'-') ' with (xxix)
- 11 -
B1n = [1 (m +q -2) (m +q-3) (m+t q-3) +(m -1) (m-2) (m+t q-1) ]lh . 2(2m+q-4) (m+t q - 1) (m+tq-3)
• [ (2m+ q -2)(2m+q -4)m] liz (xxx) m+q-3
Since B'ln > m, an exact calculation of the integral on the right hand side of (xxviii) will
yield an inequality which is less accurate than the inequality (xxi) of Liu and Martens.
Hence equality in (xxviii) will never be attained. This leads to the following observation. In his thesis [Lil, p.104], Liu proves that the function ro 1-7 rokh(ro), defined on the
sphere Sq-I , extends to a hannonic polynomial in the space H'ln-l 6) H'ln+l' i.e. there exist
polynomials Pm-l E H'ln-l and Pm+l E H'ln+l such that
Pm-l (ro) + Pm+l (ro) = rok hero) • ro E sq-l. (xxxi)
Since equality in (xxviii) will not occur it follows that neither Pm-l nor Pm+l is identically equal to zero.
- 12-
References
[Be] Berkel, C.A.M. van, Gel'fand-Shilov's function spaces of type S for several independent variables. EUT -Report, to appear.
fCZ] Calderon, A.P. and A. Zygmund, Singular integral operators and differential equations, Amer. J. Math., 79 (1957) 901-921.
lLi1] Liu. G.-Z., Evolution equations and scales of Banach spaces, thesis, June '89, Eindhoven.
[Li2] Liu, G.-Z., Hilbert spaces of harmonic functions in which differentiation operators are continuous or compact, RANA 89-09, June 1989.
lLM] Liu, G.-Z. and F.lL. Martens, Improvement of an estimate of spherical harmonics, RANA 89-10, June 1989.
[Ma] Martens, F.lL., Function spaces of harmonic and analytic functions in infinitely many variables, EUT-Report, 87-Wsk-02.
[MOS] Magnus, W., F. Oberhettinger and R.P. Soni, Formulas and theorems for mathematical physics, Springer, Berlin, 1966.
[Mii [ Miiller, C., Spherical harmonics, LNM 17 , Springer-Verlag, Berlin-Heidelberg, New-York, 1966.
[SeJ Seeley, R.T., Spherical harmonics, Amer. Math. Monthly, 73 (1966) 115-121.
[St] Stein, E.M., Singular integrals and differentiability properties of functions, Princeton University Press, Princeton, New Yersey, 1970.
Top Related