Elements of Abstract Harmonic Analysis

270 Pages·1964·2.69 MB·English

by  George Bachman

Elements of
Abstract Harmonic
Analysis
By George Bachman
POLYTECHNIC INSTITUTE OF BROOKLYN
DEPARTMENT OF MATHEMATICS
BROOKLYN, NEW YORK
with the assistance of
Lawrence Narici
POLYTECHNIC INSTITUTE OF BROOKLYN
DEPARTMENT OF MATHEMATICS
BROOKLYN, NEW YORK
ACADEMIC PRESS New York and London
COPYRIGHT © 1964, BY ACADEMIC PRESS INC.
ALL RIGHTS RESERVED.
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PRINTED IN THE UNITED STATES OP AMERICA
Preface
AbstractHarmonic Aualysis is an active branch of modern
analysis which is increasing in importance as a standard
course for the beginning graduate student. Concepts like
Banach algebras, Haar measure, locally compact Abelian
groups, etc., appear in many current research papers. This
book is intended to enable the student to approach the
original literature more quickly by informing him of these
concepts and the basic theorems involving them.
In order to give a reasonably complete and self-contained
introduction to the subject, most of the proofs have been
presented in great detail thereby making the development
understandable to a very wide audience. Exercises have been
supplied at the end of each chapter. Some of these are meant
to extend the theory slightly while others should serve to
test the reader's understanding of the material presented.
The first chapter and part of the second give a brief
review of classical Fourier analysis and present concepts
which will subsequently be generalized to a more abstract
framework. The presentation of this material is not meant
to be detailed but is given mainly to motivate the generaliza-
tions obtained later in the book. The next five chapters pre-
sent an introduction to commutative Banach algebras, gen-
eral topological spaces, and topological groups. We hope that
Chapters 2—6 might serve as an adequate introduction for
those students primarily interested in the theory of com-
mutative Banach algebras as well as serving as needed pre-
requisite material for the abstract harmonic analysis. The
remaining chapters contain some of the measure theoretic
background, including the Haar integral, and an extension of
the concepts of the first two chapters to Fourier analysis on
locally compact topological abelian groups.
In an attempt to make the book as self-contained and as
introductory as possible, it was felt advisable to start from
scratch with many concepts—in particular with general
V
vi Preface
topological spaces. However, within the space limitations,
it was not possible to do this with certain other background
material—notably some measure theory and a few facts from
functional analysis. Nevertheless, the material needed from
these areas has all been listed in various appendices to the
chapters to which they are most relevant.
There are now a number of more advanced books on
abstract harmonic analysis which go deeper into the subject.
We cite in particular the references to Rudin, Loomis, and
the recent book by Hewitt and Ross. Our treatment of the
latter part of Chapter 12 follows to some extent the initial
chapter of the book by Rudin, which would be an excellent
continuation for the reader who wishes to pursue these
matters further.
The present book is based on a one semester course in
abstract harmonic analysis given at the Polytechnic In-
stitute of Brooklyn during the summer of 1963, for which
lecture notes were written by Lawrence Narici. A few re-
visions and expansions have been made.
I would like to express my sincere gratitude to Mr. Narici
for his effort in writing the notes, improving many of the'
proofs, and for editing the entire manuscript. I would like
finally to express also my appreciation to Melvin Maron for
his help in the preparation of the manuscript.
GEORGE BACHMAN
Contents
PREFACE V
SYMBOLS USED IN TEXT x
Chapter 1 The Fourier Transform on the Real Line for
Functions in L1
1
Introduction
1
Notation 1
The Fourier Transform 2
Recovery 4
Relation between the Norms of the Fourier Transform and the
Function 10
Appendix to Chapter 1 15
Exercises 17
References 18
Chapter 2 The Fourier Transform on the Real Line for
Functions in L2 19
Inversion inL2 21
Normed and Banach Algebras 25
Analytic Properties of Functions from C intoBtrnach Algebras 29
Exercise 33
References 33
Chapter 3 Regular Points and Spectrum 34
Compactness of the Spectrum 38
Introduction to the Gel'fand Theory of Commutative Banach
Algebras 48
The Quotient Algebra 50
Exercises 53
References 54
Chapter 4 More on the Gel'fand Theory and an Intro-
duction to Point Set Topology 55
Topology 60
A Topological Space 61
vii
vii Contents
Examples of Topological Spaces 61
Further Topological Notions 62
The Neighborhood Approach 66
Exercises 71
References 72
Chapter 5 Further Topological Notions 73
Bases, Fundamental Systems of Neighborhoods, and 73
The Relative Topology and Product Spaces 78
Separation Axioms and Compactness 79
The Tychonoff Theorem and Locally Compact Spaces 84
A Neighborhood Topology for the Set of Maximal Ideals over a
Banach Algebra 87
Exercises 89
References 90
Chapter 6 Compactness of the Space of Maximal Ideals
over a Banach Algebra; an Introduction to
Topological Groups and Star Algebras 91
Star Algebras 97
Topological Groups 98
Exercises 106
References 106
Chapter 7 The Quotient Group of a Topological Group
and Some Further Topological Notions 107
Locally Compact Topological Groups 107
Subgroups and Quotient Groups 109
Directed Sets and Generalized Sequences 116
Further Topological Notions 117
Exercises 123
References 124
Chapter 8 Right Haar Measures and the Hoar Covering
Function 125
Notation and Some Measure Theoretic Results 12$
The Haar Covering Function 129
Summary of Theorems in Chapter 8 147
Exercises 149
References 149
Contents ix
Chapter 9 The Existence of a Right Invariant Haar
Integral over any Locally Compact Topolog-
ical Group 150
The Daniell Extension Approach 158
A Measure Theoretic Approach 160
Appendix to Chapter 9 163
Exercises 164
References 165
Chapter 10 The Daniell Extension from a Topological
Point of View, Some General Results from
Measure Theory, and Group Algebras 166
Extending the Integral 166
Uniqueness of the Integral 169
Examples of Haar Measures 172
Product Measurea 176
Exercises 186
References 187
Chapter 11 Characters and the Dual Group of a Locally
Compact, Abelian, Topological Group 188
Charactersand the Dual Group 192
Examples of Characters 200
Exercises 206
References 206
Chapter 12 Generalization of the Fourier Transform to
L1(G) and L2(G) 207
TheFourier Transform on L1(G) 207
Complex Measures 211
The Fourier-Stieltjes Transform 219
Positive Definite Functions 220
The Fourier Transform on L2(G) 235
Exercises 243
Appendix to Chapter 12 244
References 250
BIBLIOGRAPHY 251
INDEX 253