Abstract Harmonic Analysis of Continuous Wavelet Transforms, Issue 1863

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Springer Science & Business Media, Feb 11, 2005 - Mathematics - 193 pages
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This volume contains a systematic discussion of wavelet-type inversion formulae based on group representations, and their close connection to the Plancherel formula for locally compact groups. The connection is demonstrated by the discussion of a toy example, and then employed for two purposes: Mathematically, it serves as a powerful tool, yielding existence results and criteria for inversion formulae which generalize many of the known results. Moreover, the connection provides the starting point for a – reasonably self-contained – exposition of Plancherel theory. Therefore, the volume can also be read as a problem-driven introduction to the Plancherel formula.

 

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Contents

Introduction
1
12 Overview of the Book
4
13 Preliminaries
5
Wavelet Transforms and Group Representations
15
22 Coherent States and Resolutions of the Identity
18
23 Continuous Wavelet Transforms and the Regular Representation
21
24 Discrete Series Representations
26
25 Selfadjoint Convolution Idempotents and Support Properties
39
382 Construction Details
99
Plancherel Inversion and Wavelet Transforms
105
42 Plancherel Inversion
113
43 Admissibility Criteria
119
44 Admissibility Criteria and the Type I Condition
129
45 Wigner Functions Associated to Nilpotent Lie Groups
130
Admissible Vectors for Group Extensions
139
51 Quasiregular Representations and the Dual Orbit Space
141

26 Discretized Transforms and Sampling
45
27 The Toy Example
51
The Plancherel Transform for Locally Compact Groups
59
32 Regularity Properties of Borel Spaces
66
33 Direct Integrals
67
332 Direct Integrals of von Neumann Algebras
69
34 Direct Integral Decomposition
71
342 Central Decompositions
74
343 Type I Representations and Their Decompositions
75
344 Measure Decompositions and Direct Integrals
79
35 The Plancherel Transform for Unimodular Groups
80
36 The Mackey Machine
85
37 OperatorValued Integral Kernels
93
38 The Plancherel Formula for Nonunimodular Groups
97
52 Concrete Admissibility Conditions
145
53 Concrete and Abstract Admissibility Conditions
155
54 Wavelets on Homogeneous Groups
160
55 Zak Transform Conditions for WeylHeisenberg Frames
162
Sampling Theorems for the Heisenberg Group
169
61 The Heisenberg Group and Its Lattices
171
62 Main Results
172
63 Reduction to WeylHeisenberg Systems
174
64 WeylHeisenberg Frames
176
65 Proofs of the Main Results
178
66 A Concrete Example
182
References
185
Index
191
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