Elements of Abstract Harmonic Analysis

From Preface: Abstract Harmonic Aualysis is an active branch of modern analysis which is increasing in importance as a standard course for the beginning graduate student. Concepts like Banach algebras, Haar measure, locally compact Abelian groups, etc., appear in many current research papers. This book is intended to enable the student to approach the original literature more quickly by informing him of these concepts and the basic theorems involving them. 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. Cover Elements of Abstract Harmonic Analysis COPYRIGH © 1964, BY ACADEMIC P ISBN 1483256782 ISBN: 978-1483256788 LCCN: 64-21663 Preface Contents 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 1. Fatou's Lemma 2. Lebesgue's Theorem on Dominated Convergence 3. FubIni's Theorem 4. Tonelli-Hobson Theorem 4. Cavchy-Schwarz Inequality 5. Mlnkowski's Inequality 6. Holder's Inequality Exercises REFERENCES CHAPTER 2 The Fourier Transform on the Real Line for Functions in L2 Fourier Transforms in L2 Inversion in L2 Normed and Banach Algebras Analytic Properties of Functions from C into Banach Algebras Exercise CHAPTER 3 Regular Points and Spectrum Compactness of the Spectrum Introduction to the Gel'fand 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 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 Examples of Topological Groups Exercises REFERENCES CHAPTER 7 The Quotient Group of a Topological Group and Some Further Topological Notions Locally Compact Topological Groups Subgroups and the 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 Hoar Covering Function Summary of Theorems in Chapter 8 Remarks 5—9 Inclusive 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 Transformto L(G) and L(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 Riesz Representation Theorem Radon-Nikodym Theorem Proof of Bochner's Theor REFERENCES Bibliography Index Back Cover