Elements of Abstract Harmonic Analysis - 1st Edition - ISBN: 9781483256788, 9781483267562

Elements of Abstract Harmonic Analysis

1st Edition

Authors: George Bachman
eBook ISBN: 9781483267562
Imprint: Academic Press
Published Date: 1st January 1964
Page Count: 266
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Description

Elements of Abstract Harmonic Analysis provides an introduction to the fundamental concepts and basic theorems of abstract harmonic analysis. 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 next five chapters present an introduction to commutative Banach algebras, general topological spaces, and topological groups. 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.

Table of Contents


Preface

Symbols Used in Text

Chapter 1 The Fourier Transform on the Real Line for Functions in L1

Introduction

Notation

The Fourier Transform

Recovery

Relation between the Norms of the Fourier Transform and the Function

Appendix to Chapter 1

Exercises

References

Chapter 2 The Fourier Transform on the Real Line for Functions in L2

Inversion in L2

Normed and Banach Algebras

Analytic Properties of Functions from C into Banach Algebras

Exercise

References

Chapter 3 Regular Points and Spectrum

Compactness of the Spectrum

Introduction to the GeFfand Theory of Commutative Banach Algebras

The Quotient Algebra

Exercises

References

Chapter 4 More on the Gel'fand Theory and an Introduction to Point Set Topology

Topology

A Topological Space

Examples of Topological Spaces

Further Topological Notions

The Neighborhood Approach

Exercises

References

Chapter 5 Further Topological Notions

Bases, Fundamental Systems of Neighborhoods, and Subbases

The Relative Topology and Product Spaces

Separation Axioms and Compactness

The Tychonoff Theorem and Locally Compact Spaces

A Neighborhood Topology for the Set of Maximal Ideals over a Banach Algebra

Exercises

References

Chapter 6 Compactness of the Space of Maximal Ideals over a Banach Algebra; an Introduction to Topological Groups and Star Algebras

Star Algebras

Topological Groups

Exercises

References

Chapter 7 The Quotient Group of a Topological Group and Some Further Topological Notions

Locally Compact Topological Groups

Subgroups and Quotient Groups

Directed Sets and Generalized Sequences

Further Topological Notions

Exercises

References

Chapter 8 Right Haar Measures and the Haar Covering Function

Notation and Some Measure Theoretic Results

The Haar Covering Function

Summary of Theorems in Chapter 8

Exercises

References

Chapter 9 The Existence of a Right Invariant Haar Integral over any Locally Compact Topological Group

The Daniell Extension Approach

A Measure Theoretic Approach

Appendix to Chapter 9

Exercises

References

Chapter 10 The Daniell Extension from a Topological Point of View, Some General Results from Measure Theory, and Group Algebras

Extending the Integral

Uniqueness of the Integral

Examples of Haar Measures

Product Measures

Exercises

References

Chapter 11 Characters and the Dual Group of a Locally Compact, Abelian, Topological Group

Characters and the Dual Group

Examples of Characters

Exercises

References

Chapter 12 Generalization of the Fourier Transform to L1(G) and L2(G)

The Fourier Transform on L1(G)

Complex Measures

The Fourier-Stieltjes Transform

Positive Definite Functions

The Fourier Transform on L2(G)

Exercises

Appendix to Chapter 12

References

Bibliography

Index

Details

No. of pages:
266
Language:
English
Copyright:
© Academic Press 1964
Published:
Imprint:
Academic Press
eBook ISBN:
9781483267562

About the Author

George Bachman