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.

Contenu

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