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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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


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© Academic Press 1964
Academic Press
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About the Author

George Bachman

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