Table of Contents
Preface xi
Abstract Harmonic Analysis 1
1 Duality of Finite Abelian Groups 3
1.1 Characters 3
1.2 Dual group 6
2 Harmonic Analysis on Finite Abelian Groups 9
2.1 Fourier transformation 9
2.2 Convolution 11
2.3 Convolution operators 13
3 Set Theory and Topology 17
3.1 Basics from set theory 17
3.2 Topological background 20
3.3 Separation theorems, Uryshon's Lemma 26
3.4 Compactification 31
3.5 Partition of the unity 32
3.6 Connectedness 33
3.7 Topological groups 35
3.8 Topological subgroups, factor groups 37
3.9 Topological vector spaces 40
3.10 The Minkowski functional 43
3.11 Conjugate spaces 46
3.12 The Hahn-Banach Theorem 49
3.13 The Stone-Weierstrass Theorem 55
4 Invariant Means on Abelian Groups 61
4.1 Means on Abelian groups 61
4.2 Invariant means 63
5 Duality of Discrete and Compact Abelian Groups 67
5.1 Characters on discrete Abelian groups 67
5.2 Characters on compact, Abelian groups 68
5.3 Duality of discrete Abelian groups 69
5.4 Convolution operators on compact Abelian groups 71
5.5 Duality of compact Abelian groups 75
6 Duality of Elementary Abelian Groups 79
6.1 The dual of some special Abelian groups 79
6.2 Elementary Abelian groups 81
7 Harmonic Analysis on Compact Abelian Groups 83
7.1 The Riesz Representation Theorem 83
7.2 Haar measure on the complex unit circle 86
7.3 Fourier scries 87
7.4 Fourier analysis on compact elementary Abelian groups 90
7.5 Fourier analysis on compact Abelian groups 94
7.6 Integrahle functions on compact Abelian groups 96
7.7 Translation invariant spaces 97
8 Duality of Locally Compact Abelian Groups 101
8.1 Compactly generated locally compact Abelian groups 101
8.2 The approximation theorem of compactly generated locally compact Abelian groups 104
8.3 Duality theory of locally compact Abelian groups 107
9 Haar Integral on Locally Compact Abelian Groups 111
9.1 Haar measure and Haar integral 111
9.2 The existence of Haar measure on locally compact Abelian groups 113
9.3 The uniqueness of the Haar integral 115
9.4 Convolution 116
9.5 Haar measure on elementary and on compactly generated Abelian groups 118
10 Harmonic Analysis on Locally Compact Abelian Groups 121
10.1 Harmonic analysis on the group of integers 121
10.2 Commutative algebras 122
10.3 The Fourier transform of integrable functions 125
10.4 Fourier transformation on locally compact Abelian groups 127
Spectral Analysis and Synthesis 129
11 Basic Concepts 131
11.1 Basics from ring theory 131
11.2 Vector modules 133
11.3 Vector modules, group representations, and actions 136
11.4 Spectral analysis and synthesis on vector modules 137
11.5 Varieties on groups 142
11.6 Anmhilators 144
12 Basic Function Classes 151
12.1 Exponentials L51
12.2 Modified differences 156
12.3 Automorphisms of the measure algebra 158
12.4 Generalized exponential monomials 160
12.5 Generalized polynomials 163
12.6 Generalized exponential polynomials 170
12.7 Exponential monomials and polynomials 175
12.8 Description of exponential polynomials 180
12.9 An example 184
13 The Torsion Free Rank 187
13.1 Basics from group theory 187
13.2 The torsion free rank and polynomials 192
13.3 The polynomial ring 1&4
14 Spectral Analysis 201
14.1 Review on basics 201
11.1 Spectral analysis and torsion properties 203
15 Spectral Synthesis 207
15.1 Review on basics and history 207
15.2 Synthesizable varieties 209
15.3 Spectral synthesis on Abelian groups 211
Bibliography 215
Index 221
