Fourier Analysis on Groups

285 Pages·1962·16.401 MB·English

by  Walter Rudin

FOURIER ANALYSIS
ON GROUPS
FOURIER ANALYSIS
ON GROUPS
WALTER RUDIN
University of Wisconsin, Madison, Wisconsin
Wiley Classics Library Edition Published 1990
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FIRST PUBLISHED 1962
ALL RIGHTS RESERVED
LIBRARY OF CONGRESS CATALOG CARD NUMBER 62—12211
ISBN 0-471-52364-X (pbk.)
10 9 8 7 6 5
PREFACE
In classical Fourier analysis the action takes place on the unit
circle, on the integers and on the real line. During the last 25 or 30
years, however, an increasing number of mathematicians have
adopted the point of view that the most appropriate setting for the
development of the theory of Fourier analysis is furnished by the
class of all locally compact abelian groups. The relative ease with
which the basic concepts and theorems can be transferred to this
general context may be one of the factors which contributes to the
feeling of some that this extension is a dilution of the classical
theory, that it is merely generalization for the sake of generali-
zation.
However, group-theoretic considerations seem to be inherent in
the subject. They are implicit in much of the classical work, and
their explicit introduction has led to many interesting new analytic
problems (it is one of the aims of this book to prove this point) as
well as to conceptual clarifications. To cite a very rudimentary
example: In discussing Fourier transforms on the line it helps to
have two lines in mind, one for the functions and one for their
transforms, and to realize that each is the dual group of the other.
Also, there are classical subjects which lead almost inevitably
to this extension of the theory. For instance, Bohr (1) noticed
almost 50 years ago that the unique factorization theorem for
positive integers allows us to regard every ordinary Dirichlet series
as a power series in infinitely many variables. The boundary values
yield a function of infinitely many variables, periodic in each, that
is to say, a function on the infinite-dimensional torus Γω. It then
becomes of interest to know the closed subgroups of Τω, and it
turns out that these comprise all compact metric abelian groups.
Once we agree to admit these groups we have to admit their duals,
i.e., the countable discrete abelian groups, and since the class of all
locally compact abelian groups can be built up from the compact
ones, the discrete ones, and the euclidean spaces, it would seem
M
Vi FOURIER ANALYSIS ON GROUPS
artificial to restrict ourselves to a smaller subclass.
The principal objects of study in the present book are the group
algebras V-lfi) and M(G); Σ>τ(0) consists of all complex functions
on the group G which are integrable with respect to the Haar
measure of G, M(G) consists of all bounded regular Borel measures
on G, and multiplication is defined in both cases by convolution.
Although certain aspects of these algebras have been studied f orn on-
commutative groups G, I restrict myself to the abelian case. Other
Lv- spaces appear occasionally, but are not treated systematically.
The development of the general theory, given in Chapter 1, is
based on some simple facts concerning Banach algebras; these, as
well as other background material, are collected in the Appendices
at the end of the book. It seems appropriate to develop the material
in this way, since much of the early work on Banach algebras was
stimulated by Fourier analysis. Chapter 2 contains the structure
theory of locally compact abelian groups. These two chapters are
introductory, and most of their content is well known.
The material of Chapters 3 to 9, on the other hand, has not
previously appeared in book form. Most of it is of very recent
vintage, many of the results were obtained only within the last
two or three years, and although the solutions of some of the prob-
lems under consideration are fairly complete by now, many open
questions remain.
My own work in this field has been greatly stimulated by con-
versations and correspondence with Paul J. Cohen, Edwin Hewitt,
Raphael Salem, and Antoni Zygmund, and by my collaboration
with Henry Helson, Jean-Pierre Kahane, and YitzhakKatznelson.
It is also a pleasure to thank the Alfred P. Sloan Foundation for
its generous financial support.
Madison, Wisconsin WALTER RUDIN
November 1960
CONTENTS
CHAPTER 1
The Basic Theorems of Fourier Analysis
1.1 Haar Measure and Convolution 1
1.2 The Dual Group and the Fourier Transform 6
1.3 Fourier-Stieltjes Transforms 13
1.4 Positive-Definite Functions 17
15 The Inversion Theorem 21
1.6 The Plancherel Theorem 26
1.7 The Pontryagin Duality Theorem 27
1.8 The Bohr Compactification 30
1.9 A Characterization of Β(Γ) 32
CHAPTER 2
The Structure of Locally Compact Abelian Groups
2.1 The Duality between Subgroups and Quotient Groups 35
2.2 Direct Sums 36
2.3 Monothetic Groups 39
2.4 The Principal Structure Theorem 40
2.5 The Duality between Compact and Discrete Groups 44
2.6 Local Units in Α(Γ) 48
2.7 Fourier Transforms on Subgroups and on Quotient Groups . .. 53
CHAPTER 3
Idempotent Measures
3.1 Outline of the Main Result 59
3.2 Some Trivial Cases 61
3.3 Reduction to Compact Groups 62
3.4 Decomposition into Irreducible Measures 63
3.5 Five Lemmas 66
3.6 Characterization of Irreducible Idempotents 69
3.7 Norms of Idempotent Measures 72
3.8 A Multiplier Problem 73
CHAPTER 4
Homomorphisms of Group Algebras
4.1 Outline of the Main Result 77
4.2 The Action of Piecewise Affine Maps 79
[vii]
VIII CONTENTS
4.3 Graphs in the Coset Ring 80
4.4 Compact Groups 82
4.5 The General Case 85
4.6 Complements to the Main Result 87
4.7 Special Cases 92
CHAPTER 5
Measures and Fourier Transforms on Thin Sets
5.1 Independent Sets and Kronecker Sets 97
5.2 Existence of Perfect Kronecker Sets 99
5.3 The Asymmetry of M(G) 104
5.4 Multiplicative Extension of Certain Linear Functionals 108
5.5 Transforms of Measures on Kronecker Sets 112
5.6 Helson Sets 114
5.7 Sidon Sets 120
CHAPTER 6
Functions of Fourier Transforms
6.1 Introduction 131
6.2 Sufficient Conditions 132
6.3 Range Transformations on Β(Γ) for Non-Compact Γ 135
9.4 Some Consequences 140
6.5 Range Transformations on Α(Γ) for Discrete Γ 141
6.6 Range Transformations on Α(Γ) for Non-Discrete Γ 143
6.7 Comments on the Predecing Theorems 147
6.8 Range Transformations on Some Quotient Algebras 149
6.9 Operating Functions Defined in Plane Regions 153
CHAPTER 7
Closed Ideals in V (G)
7.1 Introduction 157
7.2 Wiener's Tauberian Theorem 159
7.3 The Example of Schwartz 165
7.4 The Examples of Herz 166
7.5 Polyhedral Sets 169
7.6 Malliavin's Theorem 172
7.7 Closed Ideals Which Are Not Self-Adjoint 181
7.8 Spectral Synthesis of Bounded Functions 183
CHAPTER 8
Fourier Analysis on Ordered Groups
8.1 Ordered Groups 193
8.2 The Theorem of F. and M. Riesz 198
CONTENTS IX
8.3 Geometrie Means 203
8.4 Factorization Theorems in H1^) and in HZ{G) 205
8.5 Invariant Subspaces of H2{G) 210
8.6 A Gap Theorem of Paley 213
8.7 Conjugate Functions 216
CHAPTER 9
Closed Subalgebras of L*(G)
9.1 Compact Groups 231
9.2 Maximal Subalgebras 232
9.3 The Stone-Weierstrass Property 239
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
Fourier Analysis on Groups
by Walter Rudin
Copyright © 1962 Wiley-Interscience.
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 Cartan and Godement [1], Loomis [1],
and Weil [1].
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."
/./. 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) tn(E + x)= m(E)
for every x € 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 [1], Loomis [1], Montgo-
mery and Zippin [1], and Weil [1]. The idea of the proof is to
construct a positive translation-invariant hnear functional T on
C (G), the space of all continuous complex functions on G with
e
compact support. This means that Tf ^ 0 if / ^ 0 and that
T(f ) = Tf, where f is the translate of / defined by
x x
(2) ÍM=fÜ-x) 13/*G).
As soon as this is done, the Riesz representation theorem shows
that there is a measure m with the required properties, such that
i
2 FOURIER ANALYSIS ON GROUPS
(3) Tf = j fdm (/eC (G)).
G e
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 justi-
fied by the following uniqueness theorem:
// m and m' are two Haar measures on G, then m' — km, where λ
is a positive constant.
Proof: Fix geC (G) so that j gdm = 1. Define λ by
e
¡ g{-x)dm'{x)=X.
G
For any f€C (G) we then have
c
j fdm' = ¡ g{y)dm{y) j f(x)dm'(x)
G G G
= ¡ g{y)dfn{y) ¡ f(z + y)dm\x)
G G
= j dm'(x) ¡ g(y)f(x + y)dm{y)
G G
= l d™>'(?) j g(y - x)f(y)<*™{y)
G G
= \ 1{y)dm{y) \ g(y - x)dm,{x) = λ ¡J dm.
G G
Hence m' = Xm. Note that the use of Fubini's theorem was legi-
timate in the preceding calculation, since theintegrandsg(y)/(a;+y)
and g(y — x)f(y) are in C (G X G).
e
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 contra-
dictory 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 / f(x)dx in place of / fdm. Thus dx, dy, . . .
c G
will always denote integration with respect to Haar measure.