Elements of Abstract Harmonic Analysis

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