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Abstract Harmonic Analysis of Continuous Wavelet Transforms : Lecture Notes in Mathematics - Hartmut Fuhr

Abstract Harmonic Analysis of Continuous Wavelet Transforms

Lecture Notes in Mathematics

Paperback Published: April 2005
ISBN: 9783540242598
Number Of Pages: 193

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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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From the reviews:

"It has become evident that the one-dimensional continuous wavelet transform has a natural foundation in unitary representation theory. ... also various multidimensional generalizations as well as the windowed Fourier transform can be treated within the general context of discrete series transformations. ... The present book develops a unified theory in an even more general setting, going beyond the discrete series. ... The book is very well written and can be recommended to anyone interested in wavelet analysis and/or representation theory." (Margit Rosler, Mathematical Reviews, Issue 2006 m)

"This book deals with generalizations, valid for general locally compact groups and unitary representations on arbitrary Hilbert spaces. ... The book is well written, and it contains a nice blend of the general analysis and the concrete examples in wavelet analysis and Gabor analysis. The book is strongly recommended for readers with interest in abstract harmonic analysis, and to readers who are familiar with wavelets and want to understand them from a broader perspective." (Ole Christensen, Zentralblatt MATH, Vol. 1060, 2005)

Introductionp. 1
The Point of Departurep. 1
Overview of the Bookp. 4
Preliminariesp. 5
Wavelet Transforms and Group Representationsp. 15
Haar Measure and the Regular Representationp. 15
Coherent States and Resolutions of the Identityp. 18
Continuous Wavelet Transforms and the Regular Representationp. 21
Discrete Series Representationsp. 26
Selfadjoint Convolution Idempotents and Support Propertiesp. 39
Discretized Transforms and Samplingp. 45
The Toy Examplep. 51
The Plancherel Transform for Locally Compact Groupsp. 59
A Direct Integral View of the Toy Examplep. 59
Regularity Properties of Borel Spacesp. 66
Direct Integralsp. 67
Direct Integrals of Hilbert Spacesp. 67
Direct Integrals of von Neumann Algebrasp. 69
Direct Integral Decompositionp. 71
The Dual and Quasi-Dual of a Locally Compact Groupp. 71
Central Decompositionsp. 74
Type I Representations and Their Decompositionsp. 75
Measure Decompositions and Direct Integralsp. 79
The Plancherel Transform for Unimodular Groupsp. 80
The Mackey Machinep. 85
Operator-Valued Integral Kernelsp. 93
The Plancherel Formula for Nonunimodular Groupsp. 97
The Plancherel Theoremp. 97
Construction Detailsp. 99
Plancherel Inversion and Wavelet Transformsp. 105
Fourier Inversion and the Fourier Algebrap. 105
Plancherel Inversionp. 113
Admissibility Criteriap. 119
Admissibility Criteria and the Type I Conditionp. 129
Wigner Functions Associated to Nilpotent Lie Groupsp. 130
Admissible Vectors for Group Extensionsp. 139
Quasiregular Representations and the Dual Orbit Spacep. 141
Concrete Admissibility Conditionsp. 145
Concrete and Abstract Admissibility Conditionsp. 155
Wavelets on Homogeneous Groupsp. 160
Zak Transform Conditions for Weyl-Heisenberg Framesp. 162
Sampling Theorems for the Heisenberg Groupp. 169
The Heisenberg Group and Its Latticesp. 171
Main Resultsp. 172
Reduction to Weyl-Heisenberg Systemsp. 174
Weyl-Heisenberg Framesp. 176
Proofs of the Main Resultsp. 178
A Concrete Examplep. 182
Referencesp. 185
Indexp. 191
Table of Contents provided by Ingram. All Rights Reserved.

ISBN: 9783540242598
ISBN-10: 3540242597
Series: Lecture Notes in Mathematics
Audience: General
Format: Paperback
Language: English
Number Of Pages: 193
Published: April 2005
Publisher: Springer-Verlag Berlin and Heidelberg Gmbh & Co. Kg
Country of Publication: US
Dimensions (cm): 23.11 x 15.49  x 1.27
Weight (kg): 0.32

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