Elements of Abstract Harmonic Analysis (eBook)

Front Cover 
1 
Elements of Abstract Harmonic Analysis 4
Copyright Page 5
Table of Contents 8
Preface 6
Symbols Used in Text 11
CHAPTER 1. The Fourier Transform on the Real Line for Functions in L1 14
Introduction 14
Notation 14
The Fourier Transform 15
Recovery 17
Relation between the Norms of the Fourier Transform and the Function 23
Appendix to Chapter 1 28
Exercises 30
REFERENCES 31
CHAPTER 2. The Fourier Transform on the Real Line for Functions in L2 32
Fourier Transforms in L2 32
Inversion in L2 34
Normed and Banach Algebras 38
Analytic Properties of Functions from C into Banach Algebras 42
Exercise 46
REFERENCES 46
CHAPTER 3. Regular Points and Spectrum 47
Compactness of the Spectrum 51
Introduction to the Gel'fand Theory of Commutative Banach Algebras 61
The Quotient Algebra 63
Exercises 66
REFERENCES 67
CHAPTER 4. More on the Gel'fand Theory and an Introduction to Point Set Topology 68
Topology 73
A Topological Space 74
Examples of Topological Spaces 74
Further Topological Notions 75
The Neighborhood Approach 79
Exercises 84
REFERENCES 85
CHAPTER 5. Further Topological Notions 86
Bases, Fundamental Systems of Neighborhoods, and Subbases 86
The Relative Topology and Product Spaces 91
Separation Axioms and Compactness 92
The Tychonoff Theorem and Locally Compact Spaces 97
A Neighborhood Topology for the Set of Maximal Ideals over a Banach Algebra 100
Exercises 102
REFERENCES 103
CHAPTER 6. Compactness of the Space of Maximal Ideals over a Banach Algebra an Introduction to Topological Groups and Star Algebras
Star Algebras 110
Topological Groups 111
Exercises 119
REFERENCES 119
CHAPTER 7. The Quotient Group of a Topological Group and Some Further Topological Notions 120
Locally Compact Topological Groups 120
Subgroups and the Quotient Groups 122
Directed Sets and Generalized Sequences 129
Further Topological Notions 130
Exercises 136
REFERENCES 137
CHAPTER 8. Right Haar Measures and the Haar Covering Function 138
Notation and Some Measure Theoretic Results 138
The Haar Covering Function 142
Summary of Theorems in Chapter 8 160
Exercises 162
REFERENCES 162
CHAPTER 9. The Existence of a Right Invariant Haar Integral over any Locally Compact Topological Group 163
The Daniell Extension Approach 171
A Measure Theoretic Approach 173
Appendix to Chapter 9 176
Exercises 177
REFERENCES 178
CHAPTER 10. The Daniell Extension from a Topological Point of View, Some General Results from Measure Theory, and Group Algebras 179
Extending the Integral 179
Uniqueness of the Integral 182
Examples of Haar Measures 185
Product Measures 189
Exercises 199
REFERENCES 200
CHAPTER 11. Characters and the Dual Group of a Locally Compact, Abelian, Topological Group 201
Characters and the Dual Group 205
Examples of Characters 213
Exercises 219
REFERENCES 219
CHAPTER 12. Generalization of the Fourier Transform to L1(G) and L2(G) 220
The Fourier Transform on L1(G) 220
Complex Measures 224
The Fourier-Stieltjes Transform 232
Positive Definite Functions 233
The Fourier Transform on L2(G) 248
Exercises 256
Appendix to Chapter 12 257
REFERENCES 263
Bibliography 264
Index 266