Fourier Analysis.pdf

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FOURIER ANALYSIS ON FINITE GROUPS (AND OTHER NICEGROUPS)

MARK H. KIM

1. Introduction

One of the most useful tools in mathematical analysis is the Fourier series ofa periodic function, which decomposes the given function into a sum of simpleoscillatory functions. The classical case deals with an L-periodic function f on R

and its Fourier series

n=

f(n)e2inx/L,

where the nth Fourier coefficient f(n) is defined to be

(1) f(n) =1

L

L/2L/2

f(x)e2inx/L dx

for each n Z. An elegant theory emerges when we restrict our attention tosquare-integrable functions on [L/2, L/2], the collection of which is a Hilbertspace, denoted by L2([L/2, L/2]), with the inner product

f, gL2([L/2,L/2]) =1

LL/2L/2

f(x)g(x) dx

and the associated L2 norm

fL2([L/2,L/2]) =

1

L

L/2L/2

|f(x)|2 dx

1/2.

Indeed, it can be shown that the Fourier series of an arbitrary square-integrablefunction converges to the function in the L2 norm, and that the Fourier basis{e2inx/L : n Z} is, in a precise sense, better than any other orthonormal (Hilbert)basis in L2([L/2, L/2]).

The classical theory of Fourier analysis also deals with nonperiodic functions,which can be thought of as the limiting case of L-periodic functions as L tends to

infinity. Indeed, we make a simple change of variable in (1) to obtain

f(n) =

1/21/2

f(Lx)e2inx dx.

By sending L to infinity, we are led to the following integral:

f(n) =

f(x)e2inx dx.

1


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2 MARK H. KIM

Since the integral makes sense even when n is not an integer, we replace n with areal variable :

f() =

f(x)e

2ix dx.

This is the Fourier transform of a complex-valued function f on R, which makessense if f is integrable on R. The L2 theory extends to the nonperiodic case aswell, with the inner product

f, gL2(R) =

f(x)g(x) dx

and the associated L2 norm

fL2(R) =

|f(x)|2 dx

1/2.

With this setting, it can be shown that the integral given by the Fourier inversionformula

f()e2ix d

converges to f in the L2 norm.Now, we might wonder in what sense the theory of Fourier series is the same

as the theory of Fourier transform on R. The analogy becomes clearer once weconsider L-periodic functions on R as functions on the circle group

T = {z C : |z| = 1}

via the change-of-variable map x e2ix/L. Indeed, both T and R are topologicalgroups, viz., the group operation and the inverse operation are continuous undertheir topologies. Furthermore, the group is abelian, and its topology is locallycompact. We also observe that integration on each group is translation-invariant,

in the sense that

f(x + h) dx =

f(x) dx

for any real number h. In other words, both groups are equipped with Haar mea-surestranslation-invariant Borel regular measures. Perhaps there is a generaltheory of Fourier analysis on such groups.

2. Fourier Analysis on Finite Cyclic Groups

The goal of the talk is to sketch the theory of Fourier analysis on locally com-pact abelian groups equipped with Haar measures. This, of course, sounds awfullycomplicated, so we make some simplifying assumptions. Indeed, we shall focus onfinite groups for the majority of the talk.

What benefit might this bring? For one, any topology on a finite group that

is compatible with the group structure must be compact and discrete. Therefore,every function on the group is continuous, every convergent sequence is eventuallyconstant, every real-valued function attains a maximum and a minimum, and soon. In fact, we may as well forget about the topology entirely, as it is as nice as itcould possibly be.

Furthermore, any nontrivial measure must assign a positive measure to eachsingleton, and the translation invariance forces every singleton to be of the samemeasure. Therefore, every function on a finite group equipped a Haar measure is


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FOURIER ANALYSIS ON FINITE GROUPS (AND OTHER NICE GROUPS) 3

integrable, and the measure of a subset of the group is little more than a constantmultiple of the cardinality of the subset. This drop measure and integration theoryfrom our discussion. The sticky technical issues of real analysis disappear entirely,thus allowing us to focus on the big picture.

We shall begin by considering cyclic groups, for the unit circle, on which thetheory of Fourier series is developed, can be approximated by regular N-gons asN tends to infinity. Since the cyclic group of order N is isomorphic to the addi-tive group Z/NZ of integers modulo N, we shall use the notations from modulararithmetic. Without further ado, we define the function space we shall be workingon:

Definition 1. The Lebesgue space on the cyclic group of order N, denoted byL(Z/NZ), is the collection of all complex-valued functions on Z/NZ. The innerproduct on L(Z/NZ) is defined to be

f, g =1

N

nZ/NZ

f(n)g(n)

for each f, g L(Z/NZ), and the associated norm is defined to be

f =

1

N

nZ/NZ

|f(n)|2

1/2

for each f L(Z/NZ).

It is easy to see that L(Z/NZ) is an N-dimensional complex vector space, andthat , is an inner product.

To develop a theory of Fourier analysis on Z/NZ, we need the analogues ofexponential functions in L(Z/NZ). Imitating the usual Fourier basis in L2(T), we

defineem(n) = e

2imn/N

for each m and every n in Z/NZ. The expression is well-defined, for e2iN/N = 1.

Proposition 2 (Orthogonality relations). e1, . . . , eN is an orthonormal basis ofL(Z/NZ).

Proof. We first observe that

em, em =1

N

nZ/NZ

1 = 1

for each m. To show that em1 , em2 = 0 for any m1 = m2, we recall that the sum

of all nth roots of unity is 0. Indeed, if n is the primitive nth root of unity, then

0 = nn 1 = (n 1)n1k=0

kn.

Since n = 1, it follows thatn1k=0

kn = 0.


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4 MARK H. KIM

We now fix two distinct elements m1, m2 Z/NZ and define m to be the greatestcommon divisor of m1 m2 and N. Setting M = N/m, we see that a simplerescaling argument yields

em1 , em2 =1

N

nZ/NZ

e2i(m1m2)n/N =1

N

nZ/NZ

e2in/M.

The latter sum consists of m copies of the sum of all Mth roots of unity, whence itequals zero. This completes the proof.

Imitating the usual definition once again, we define the cyclic analogue of theFourier series:

Definition 3. The discrete Fourier transform of f L(Z/NZ) is the function

f L(Z/NZ) defined at each m Z/NZ to be

f(m) = f, em.

The discrete Fourier series of f is defined to be the finite sum

(2)

mZ/NZ

f(m)em.

It is a standard result in linear algebra that (2) is the correct expansion in theorthonormal basis e1, . . . , eN. Therefore, we have the Fourier inversion formula

f(n) =

mZ/NZ

f(m)em(n)

for all f L(Z/NZ). Therefore, the discrete Fourier transform operator F :L(Z/NZ) L(Z/NZ) defined by the mapping

f f

is invertible. Clearly, F is a linear transformation, whence F is a linear automor-phism on L(Z/NZ). In this way, the discrete Fourier transform is similar to thenonperiodic Fourier transform, which transforms functions on R to functions on R.

Pushing the analogy further, we now establish the following theorem, whosecontinuous analogue is a cornerstone of continuous Fourier analysis:

Theorem 4 (Plancherels theorem). F is an isometry1 on L(Z/NZ), viz.,

f = N1/2f

for all f L(Z/NZ).

1We could, of course, renormalize the inner product or the Fourier transform to turn this into

an isometry. Doing so would result in the loss of orthonormal basis status of the Fourier basis,which we prefer to have in our exposition.


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Proof. By the Fourier inversion formula, we have

f2 = f, f

=

mZ/NZ

f(m)em,

mZ/NZ

f(m)em

=

mZ/NZ

mZ/NZ

f(m)f(m)em, e

m

= N 1

N

mZ/NZ

|f(m)|2

= Nf2,

as was to be shown.

Just as in the continuous case, the discrete Fourier transform is translation-

invariant. The trivial proof of the following proposition is omitted.

Proposition 5. For each f L(Z/NZ) and n0 Z/NZ, we define the translationoperator Tn0 by setting

Tn0f(n) = f(n n0).

With this definition, we have

Tn0f(m) = e2in0m/Nf(m).

3. Application: The Discrete Isoperimetric Inequality

Having developed basic Fourier analysis on Z/NZ, we now consider an applica-tion. Recall the isoperimetric inequality for curves in the plane, which states that

every rectifiable closed simple curve in the plane with length L and interior area Asatisfies the inequality

A L2

4,

whose equality is attained if and only if the curve is a circle. Our goal is to provethe following discrete analogue:

Theorem 6 (Discrete isoperimetric inequality). If z : Z/NZ C defines a closedequilateral polygon{z(0), z(1), . . . , z(N1)} centered at the origin in the plane withlength L and interior area A, then

A

L2

1

4Ncot

N

To this end, we first establish the following formula:

Lemma 7. If z is defined as in Theorem 6, then

A =N

2

N1m=0

|z(m)|2 sin2m

N.

Proof. For each 0 n N 1, we write

z(n) = x(n) + iy(n),


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6 MARK H. KIM

where x and y are real-valued functions on Z/NZ. We observe that the area of thetriangle Oz(n)z(n + 1) can be written as

12

[x(n)y(n + 1) x(n + 1)y(n)] = 12

Im (z(n)z(n + 1)) .

Since the area of the polygon is the sum of these N triangles, we have

(3) A =1

2

N1n=0

Im (z(n)z(n + 1)) =1

4i

N1n=0

z(n)z(n + 1) z(n)z(n + 1).

The Fourier inversion formula yields

z(n) =

N1m=0

z(m)em(n)

z(n) =N1

m=0

z(m)em(n)

z(n + 1) =N1m=0

e2im/Nz(m)em(n)

z(n + 1) =

N1m=0

e2im/Nz(m)em(n).

Therefore, the orthogonality relations and (3) imply

A =N

4i

N1m=0

|z(m)|2(e2im/N e2im/N) =N

2

N1m=0

|z(m)|2 sin2m

N,

as was to be shown.

We now proceed to prove the discrete isoperimetric inequality.

Proof of Theorem 6. Since the polygon is equilateral,

(4)L2

N= N

L

N

2= N |z(1) z(0)|2 =

N1n=0

|z(n + 1) z(n)|2.

By Plancherels theorem and Proposition 5, we have

N1n=0

|z(n + 1) z(n)|2 = NN1m=0

|z(m + 1) z(m)|2

= NN1

m=0

|e2im/N 1|2|z(m)|2

= 4NN1m=0

sin mk

2 |z(m)|2,whence (4) implies that

(5) L2 = (2N)2N1m=0

sin mk

2 |z(m)|2.


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By Lemma 7, we have

4N A tan

N =

2N2

tan

NN1m=0

|z(m)|2

sin2m

N

= 4N2N1m=0

|z(m)|2 sinm

N

cos

mN

tan

N

.

Therefore,

L2 4N A tan

N= 4N2

N1m=0

|z(m)|2 sinm

N

sinm

N

cos

mN

tan

N

.

We claim that L2 4N A tan N 0, which then establishes the inequality

A

L2

1

4Ncot

N.

To show this, it suffices to prove

sinm

N

sinm

N

cos

mN

tan

N

0

for each 0 m N 1. Furthermore, we have sin(m/N) 0 for all such m,whence it is enough to establish the inequality

sinm

N

cos

mN

tan

N

0,

But the above is equivalent to

tanm

N tan

N,

which holds for all such m. Therefore, the claim holds, and the proof is nowcomplete.

We remark that the discrete isoperimetric inequality achieves the equality if andonly if the polygon at hand is regular.

4. Concluding Remarks: Abstract Harmonic Analysis

We have alluded to the theory of Fourier analysis on locally compact abeliangroups with Haar measures. This is the subject of abstract harmonic analysis,which we now touch upon briefly.

Recall that a topological group is a group G with a topology such that the group-

operation map (x, y) xy on GG into G and the inverse map x x1

on G intoitself are continuous. A basic theorem in abstract harmonic analysis guarantees thatevery locally compact Hausdorff abelian group G admits a Haar measure, which isa nonzero Borel measure on G such that

(1) (E) = (gE) = (Eg) for each g G and every Borel set E G.(2) For each Borel subset E of G, the measure of E is the supremum of the

measure of compact subsets of E;(3) Every point in G has a neighborhood of finite measure.


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8 MARK H. KIM

Furthermore, all Haar measures on G are constant multiples of one another, whencewe can speak of the Haar measure on G, by stipulating the measure of the wholegroup to be 1.

How might we develop Fourier analysis on G? Going back to finite cyclicgroups, we observe that each em(n) is a group homomorphism on Z/NZ into thecircle group T. Therefore, we can consider the discrete Fourier transform as an actof describing functions on Z/NZ in terms of characters on Z/NZ. Therefore, weshould expect that Fourier analysis on G requires us to study the characters on G.

Formally, a unitary character on G is a topological group morphisma grouphomomorphism that is continuouson G into T. The collection G of unitary char-acters on G forms a group, called the dual group of G. Generalizing the notion ofrewriting the function in the Fourier space from continuous Fourier analysis, weshould expect the Fourier transform on G to transform a function on G to a func-tion on G: G shall be, in some sense, our Fourier basis. Indeed, if G is compact,then G is an orthonormal set in L2(G).

The Fourier transform of f L1

(G) is defined to be

(Ff)() = f() =

f(x)(x) dx

for each G. The basic theorems of Fourier analysis, such as the Hausdorff-Younginequality, the Fourier inversion formula, and Planceherels theorem continue tohold in this setting. Furthermore, the Pontrayagin duality theorem states that the

double dualG of G is isomorphic to G, whereby we recover the standard duality

relation between the physical space and the Fourier space.Abstract harmonic analysis is at the intersection of many disciplines of math-

ematics. To see this, let us examine the structure of G in greater detail. Thenatural topology on G is the compact-open topology. This coincides with the weak-* topology on G, which is the weak topology induced by the canonical embedding

G G. Under this topology, G is a locally compact topological group. Further-more, the Fourier transform F can be thought of as an operator on L1(G) into the

space C(G) Viewing L1(G) as a *-algebra with convolution as the product, we canprove that F is the Gelfand transform on L1(G). Indeed, F is a norm-decreasing*-homomorphism into the sapce C0(G) of continuous maps on G vanishing at infin-

ity. Pursuing this viewpoint further, we can identify G with the spectrum ofL1(G)via the *-representation

(f) =

f(x)(x) dx,

and the closed ideals in L1(G) can be studied in great detail.We can also consider unitary representations ofG, which are homomorphisms

from G into the group U(H) of unitary operators on nonempty Hilbert space H

that are continuous with respect to the strong operator topology. In this setting,H is called the representation space of , and dimH the degree of . A topic ofinterest in representation theory is the study of irreducible representations, whichis defined as follows:

(a) A closed subspace M ofH is invariant if (x)M M for all x G.(b) The restriction of on an invariant subspace is a subrepresentation of . The

subrepresentation is proper if it is a restriction on a proper subspace.(c) is irreducible if is not reducible.


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It can be shown that all irreducible representations of G are one-dimensional,whence G is the collection of irreducible representations of G. Therefore, therepresentation theory of locally compact abelian group is, in a sense, generalizedharmonic analysis. This also provides connections between harmonic analysis anddifferential geometry, where Lie groups play an important role.

References

[Fol95] Gerald B. Folland, A course on abstract harmonic analysis, CRC Press, 1995.[Sch50] I. J. Schoenberg, The finite fourier series and elementary geometry, The American Math-

ematical Monthly 57 (1950), 390404.[SS03] Elias M. Stein and Rami Shakarchi, Fourier analysis: An introduction, Princeton Univer-

sity Press, 2003.

[Ter99] Audrey Terras, Fourier analysis on finite groups and applications, Cambridge UniversityPress, 1999.

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