Fourier Analysis on Groups

by Walter Rudin

Overview

In the late 1950s, many of the more refined aspects of Fourier analysis were transferred from their original settings (the unit circle, the integers, the real line) to arbitrary locally compact abelian (LCA) groups. Rudin's book, published in 1962, was the first to give a systematic account of these developments and has come to be regarded as a classic in the field. The basic facts concerning Fourier analysis and the structure of LCA groups are proved in the opening chapters, in order to make the treatment relatively self-contained.

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ISBN-13:	9780486821016
Publisher:	Dover Publications
Publication date:	04/19/2017
Series:	Dover Books on Mathematics
Sold by:	Barnes & Noble
Format:	eBook
Pages:	304
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About the Author

Walter Rudin was an Austrian-American mathematician and professor of Mathematics at the University of Wisconsin-Madison.

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Fourier Analysis on Groups

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By Walter Rudin

Dover Publications, Inc.

Copyright © 2017 Walter Rudin
All rights reserved.
ISBN: 978-0-486-82101-6

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Contents

Chapter 1 The Basic Theorems of Fourier Analysis,
Chapter 2 The Structure of Locally Compact Abelian Groups,
Chapter 3 Idempotent Measures,
Chapter 4 Homomorphisms of Group Algebras,
Chapter 5 Measures and Fourier Transforms on Thin Sets,
Chapter 6 Functions of Fourier Transforms,
Chapter 7 Closed Ideals in L1 (G),
Chapter 8 Fourier Analysis on Ordered Groups,
Chapter 9 Closed Subalgebras of L1 (G),
Appendices,
A. Topology, 247,
B. Topological Groups, 252,
C. Banach Spaces, 256,
D. Banach Algebras, 261,
E. Measure Theory, 264,
Bibliography, 271,
List of Special Symbols, 281,
Index, 283,


CHAPTER 1

The Basic Theorems of Fourier Analysis

The material contained in this chapter forms the core of our subject and is used throughout the later part of this book. Various approaches are possible; the same subject matter is treated, from different points of view, in Cart an and Godement, Loomis, and Weil.

Unless the contrary is explicitly stated, any group mentioned in this book will be abelian and locally compact, with addition as group operation and 0 as identity element (see Appendix B). The abbreviation LCA will be used for "locally compact abelian."

1.1. Haar Measure and Convolution

1.1.1. On every LCA group G there exists a non-negative regular measure m (see Appendix E), the so-called Haar measure of G, which is not identically 0 and which is translation-invariant. That is to say,

(1) m(E + x) = m(E)

for every x [member of] G and every Borel set E in G.

For the construction of such a measure, we refer to any of the following standard treatises: Halmos, Loomis, Montgomery and Zippin, and Weil. The idea of the proof is to construct a positive translation-invariant linear functional T on Cc(G), the space of all continuous complex functions on G with compact support. This means that Tf ≥ 0 if f ≥ 0 and that T(fx) = Tf, where fx is the translate of f defined by

(2) fx(y) = f{y-x) (y [member of] G).

As soon as this is done, the Riesz representation theorem shows that there is a measure m with the required properties, such that

(3) [MATHEMATICAL EXPRESSION OMITTED].

1.1.2.If V is a non-empty open subset of G, then m(V) > 0. For if m(V) = 0 and K is compact, finitely many translates of V cover K, and hence m(K) = 0. The regularity of m then implies that m(E) = 0 for all Borel sets E in G, a contradiction.

1.1.3. We have spoken of the Haar measure of G. This is justified by the following uniqueness theorem:

If m and m' are two Haar measures on G} then m' = λm, where λ is a positive constant.

Proof: Fix g [member of] Cc(G) so that [MATHEMATICAL EXPRESSION OMITTED]. Define λ by

[MATHEMATICAL EXPRESSION OMITTED]

For any f [member of] Cc(G) we then have

[MATHEMATICAL EXPRESSION OMITTED]

Hence m' = λm. Note that the use of Fubini's theorem was legitimate in the preceding calculation, since the integrands g(y)f (x+y) and g(y — x)f(y) are in Cc(G x G).

Thus Haar measure is unique, up to a multiplicative positive constant. If G is compact, it is customary to normalize m so that m(G) = 1. If G is discrete, any set consisting of a single point is assigned the measure 1. These requirements are of course contradictory if G is a finite group, but this will cause us no difficulty.

Having established the uniqueness of m, we shall now change our notation, and write ∫G f(x)dx in place of ∫G fdm. Thus dx, dy, ... will always denote integration with respect to Haar measure.

1.1.4.For any Borel set E in G, m(- E) = m(E). For if we set m'(E) = m(- E), m' is a Haar measure on G, and so there is a constant λ such that m(- E) = λm(E) for all Borel sets E. Taking E so that - E = E, we see that λ = 1.

1.1.5. Translation inLP(G). If G is a LCA group and 1 ≤ p ≤ ∞, we shall write Lp(G) in place of Lp(m) (see Appendix E7). It is clear that the Lp-norms are translation invariant, i.e., that

(1) fx p = f p (x [member of] G),

where, we recall, fx is the translate of f defined by

(2) fx(y) = f{y-x) (y [member of] G)

Theorem. Suppose 1 ≤ p< ∞ and f [member of] Lp{G). The map

(3) x -> fx

is a uniformly continuous map of G into Lp(G).

Proof: Let ε > 0 be given. Since Cc(G) is dense in Lp(G) (Appendix E8) there exists g [member of] Cc(G), with compact support K, such that g - f p< ε/3, and the uniform continuity of g (Appendix B9) implies that there is a neighborhood V of 0 in G such that

(4) [MATHEMATICAL EXPRESSION OMITTED]

for all x [member of] V. Hence g - gx p< ε/3, and so

[MATHEMATICAL EXPRESSION OMITTED]

if x [member of] V. Finally, fx - fv = (f - fv-z)x, so that fx - fy p< ε if y - x [member of] V, and the proof is complete.

Note that the same theorem (with the same proof) is true with C0(G) in place of Lp(G), but that it is false for L∞(G), unless G is discrete.

1.1.6. Convolutions. For any pair of Borel functions f and g on the LCA group G we define their convolution f * g by the formula

(1) (f * g)(x) = ∫G f(x - y)g(y)dy

provided that

(2) ∫G|f(x - y)g(y)|dy< ∞.

Note that the integral (1) can also be written in the form

(3) ∫Gfv(x)g(y)dy

so that f * g may be regarded as a limit of linear combinations of translates of f; this statement may be made precise, but we assign it only heuristic value at present. (See Theorem 7.1.2.)

Theorem, (a) If (2) holds for some x [member of] G, then (f * g)(x) = (g * f)(x).

(b) If f [member of] L1(G) and g [member of] L∞(G), then f * g is bounded and uniformly continuous.

(c) If f and g are in Cc(G), with compact supports A and B, then the support of f * g lies in A + B, so that f * g [member of] Cc(G).

(d) If 1 < p< ∞, 1/p + 1/q = 1, f [member of] Lp(G), and g [member of] Lq(G), then f * g [member of] C0(G).

(e) If f and g are in L1(G), then (2) holds for almost all x [member of] G, f * g [member of] L1(G), and the inequality

f * g 1 ≤ f 1 g 1

holds.

(f) If f, g, h are in L1(G), then (f * g) * h = f * (g * h).

Proof: Replacing y by y + x in (1) and applying 1.1.4, we obtain

[MATHEMATICAL EXPRESSION OMITTED]

and (a) is proved.

Under the hypotheses of (b), it is clear that

|(f * g)(x)| ≤ f 1 g ∞ (x [member of] G)

so that f * g is bounded. For x [member of] G, z [member of] G, we have

[MATHEMATICAL EXPRESSION OMITTED]

Theorem 1.1.5 shows that the last expression can be made arbitrarily small by restricting x - z to lie in a suitably chosen neighborhood of 0 and (b) follows.

If f vanishes outside A and g vanishes outside B, then f(x-y)g(y) = 0 unless y [member of] B and x - y [member of] A, i.e., unless x [member of] A + B. Thus f * g vanishes outside A + B, and (c) is proved.

To prove (d), choose sequences {fn} and {gn} in Cc(G) such that fn - f p -> 0 and gn - g q 0 as n -> ∞. Hölder's inequality shows that fn * gn ->f * g uniformly. By (c), fn * gn [member of] Cc(G). Hence f * g [member of] C0(G), and (d) follows.

The proof of (e) will depend on Fubini's theorem, and we first have to show that the integrand in (1) is a Borel function on G x G. Fix an open set V in the plane, put E = f-1(V), E' = E x G, and let E" = {(x, y) : x - y [member of] E}. Then E' is a Borel set in G x G, and since the homeomorphism of G x G onto itself which carries (x, y) to (x + y, y) maps E' onto E", E" is also a Borel set. Since f(x - y) e V if and only if (x, y) e E", we see that f(x - y) is a Borel function on G x G, and so is the product f(x - y)g(y).

By Fubini's theorem,

∫G∫G|f(x - y)g(y)| dxdy = f 1 g 1.

Setting φ(x) = ∫G|f(x - y)g(y)|dy, it follows that φ [member of] L1(G). In particular, φ(x) < ∞ for almost all x, and so (f * g)(x) exists for almost all x. Finally, |(f * g)(x)| ≤ φ(x), and the proof of (e) is complete.

The proof of (f) is also an application of Fubini's theorem, justified by (e) for almost all x:

[MATHEMATICAL EXPRESSION OMITTED]

1.1.7.Theorem. For any LCA group G, L1(G) is a commutative Banach algebra, if multiplication is defined by convolution. If G is discrete, L1(G) has a unit.

Proof: The first statement follows from parts (e), (f), and (a) of Theorem 1.1.6, since the distributive law holds: f * (g + h) = f * g + f * h.

If G is discrete and the Haar measure is normalized as indicated in Section 1.1.3, then

[MATHEMATICAL EXPRESSION OMITTED]

and if e(0) = 1 but e(x) - 0 for all x ≠ 0, then e [member of] L1(G) and f * e = f. Thus e is the unit of L1(G).

1.1.8. If G is not discrete, then L1(G) has no unit (see Section 1.7.3), but approximate units are always available.

Theorem. Given f [member of] L1(G) and ε > 0, there exists a neighborhood V of 0 in G with the following property: if u is a non-negative Borel function which vanishes outside V, and if ∫G(x)dx = 1, then

f - f * u 1< ε.

Proof: By Theorem 1.1.5, we can choose V so that f - fy 1 < ε for all y [member of] V. If u satisfies the hypotheses, we have

[MATHEMATICAL EXPRESSION OMITTED]

so that

[MATHEMATICAL EXPRESSION OMITTED]

1.2. The Dual Group and the Fourier Transform

1.2.1. Characters. A complex function γ on a LCA group G is called a character of G if |γ(x)| = 1 for all x [member of] G and if the functional equation

(1) γ(x + y) = γ(x)γ(y) (x, y [member of] G)

is satisfied. The set of all continuous characters of G forms a group Γ, the dual group of G, if addition is defined by

(2) (γ1 + γ2)(x) = γ1(x)γ2(x) (x [member of] G; γ1, γ2 [member of] Γ)

Throughout this book, the letter Γ will denote the dual group of the LCA group G.

In view of the duality between G and Γ which will be established in Section 1.7, it is customary to write

(3) (x, γ)

in place of γ(x). With this notation, (1) and (2) become

(4) (x + y, γ) = (x, γ)(y, γ) and (x, γ1 + γ2) = (x, γ1)(x, γ2).

It follows immediately that

(5) (0, γ) = (x, 0) = 1 (x [member of] G, γ [member of] Γ)

and

(6) (- x, γ) = (x, -γ) = (x, γ)-1 = [bar.(x, y)].

We shall presently endow Γ with a topology with respect to which Γ will itself be a LCA group. But first we identify Γ with the maximal ideal space of L1(G) (Appendix D).

1.2.2. Theorem. If γ [member of] Γ and if

(1) [MATHEMATICAL EXPRESSION OMITTED],

then the map f -> [??](γ) is a complex homomorphism of L1(G), and is not identically 0. Conversely, every non-zero complex homomorphism of L1(G) is obtained in this way, and distinct characters induce distinct homomorphisms.

Proof: Suppose f, g [member of] L1(G), and k = f * g. Then

[MATHEMATICAL EXPRESSION OMITTED]

Thus the map f -> [??](γ) is multiplicative on the Banach algebra L1(G), and since it is clearly linear, it is a homomorphism. Since |(- x, γ)| = 1, [??](γ) ≠ 0 for some f [member of] L1(G).

For the converse, suppose h is a complex homomorphism of L1(G), h ≠ 0. Then h is a bounded linear functional of norm 1 (Appendix D4), so that

(2) [MATHEMATICAL EXPRESSION OMITTED]

for some φ [member of] L∞(G) with φ ∞ = 1 (Appendix E10). If f and g are in L1(G), we have

[MATHEMATICAL EXPRESSION OMITTED]

so that

(3) h(f)φ(y) = h(fv)

for almost all y [member of] G. By Theorem 1.1.5 and the continuity of h, the right side of (3) is a continuous function on G, for each f [member of] L1(G). Choosing f so that h(f) ≠ 0, (3) shows that φ(y) coincides with a continuous function almost everywhere, and hence we may assume that φ is continuous, without affecting (2). Then (3) holds for all y [member of] G.

(Continues...)

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Table of Contents

The Basic Theorems of Fourier Analysis.
The Structure of Locally Compact Abelian Groups.
Idempotent Measures.
Homomorphisms of Group Algebras.
Measures and Fourier Transforms on Thin Sets.
Functions of Fourier Transforms.
Closed Ideals in L1(G).
Fourier Analysis on Ordered Groups.
Closed Subalgebras of L1(G).
Appendices: Topology, Topological Groups, Banach Spaces, Banach Algebras, Measure Theory.
Bibliography.
List of Special Symbols.
Index.