Elements of Abstract Harmonic Analysis

George Bachman

• Publisher: Academic Press
• Year: 2013
• Print ISBN:
  9781483256788
• eBook ISBN:
  9781483267562
• Topic: Mathématiques
• Subtopic: Géométrie

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

• Front Cover
• Elements of Abstract Harmonic Analysis
• Copyright Page
• Table of Contents
• Preface
• Symbols Used in Text
• CHAPTER 1. The Fourier Transform on the Real Line for Functions in L1
• CHAPTER 2. The Fourier Transform on the Real Line for Functions in L2
• CHAPTER 3. Regular Points and Spectrum
• CHAPTER 4. More on the Gel'fand Theory and an Introduction to Point Set Topology
• CHAPTER 5. Further Topological Notions
• CHAPTER 6. Compactness of the Space of Maximal Ideals over a Banach Algebra; an Introduction to Topological Groups and Star Algebras
• CHAPTER 7. The Quotient Group of a Topological Group and Some Further Topological Notions
• CHAPTER 8. Right Haar Measures and the Haar Covering Function
• CHAPTER 9. The Existence of a Right Invariant Haar Integral over any Locally Compact Topological Group
• CHAPTER 10. The Daniell Extension from a Topological Point of View, Some General Results from Measure Theory, and Group Algebras
• CHAPTER 11. Characters and the Dual Group of a Locally Compact, Abelian, Topological Group
• CHAPTER 12. Generalization of the Fourier Transform to L1(G) and L2(G)
• Bibliography
• Index