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.

PrefaceSymbols Used in TextChapter 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 ReferencesChapter 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 ReferencesChapter 3 Regular Points and Spectrum Compactness of the Spectrum Introduction to the GeFfand Theory of Commutative Banach Algebras The Quotient Algebra Exercises ReferencesChapter 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 ReferencesChapter 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 ReferencesChapter 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 ReferencesChapter 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 ReferencesChapter 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 ReferencesChapter 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 ReferencesChapter 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 ReferencesChapter 11 Characters and the Dual Group of a Locally Compact, Abelian, Topological Group Characters and the Dual Group Examples of Characters Exercises ReferencesChapter 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 ReferencesBibliographyIndex