Elements of Abstract Harmonic Analysis

Elements of Abstract Harmonic Analysis

1st Edition - January 1, 1964

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  • Author: George Bachman
  • eBook ISBN: 9781483267562

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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



    The Fourier Transform


    Relation between the Norms of the Fourier Transform and the Function

    Appendix to Chapter 1



    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



    Chapter 3 Regular Points and Spectrum

    Compactness of the Spectrum

    Introduction to the GeFfand Theory of Commutative Banach Algebras

    The Quotient Algebra



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


    A Topological Space

    Examples of Topological Spaces

    Further Topological Notions

    The Neighborhood Approach



    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



    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



    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



    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



    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



    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



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

    Characters and the Dual Group

    Examples of Characters



    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)


    Appendix to Chapter 12




Product details

  • No. of pages: 266
  • Language: English
  • Copyright: © Academic Press 1964
  • Published: January 1, 1964
  • Imprint: Academic Press
  • eBook ISBN: 9781483267562

About the Author

George Bachman

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