Wavelet Transforms and Localization Operators

by Wong, Man Wah

• ISBN13: 9783764367893
• ISBN10: 376436789X
• Format: Hardcover
• Copyright: 2002-11-01
• Publisher: Springer Verlag

Summary

The focus of this book is on the Schatten-von Neumann properties and the product formulas of localization operators defined in terms of infinite-dimensional and square-integrable representations of locally compact and Hausdorff groups. Wavelet transforms, which are the building blocks of localization operators, are also studied in their own right. Daubechies operators on the Weyl-Heisenberg group, localization operators on the affine group, and wavelet multipliers on the Euclidean space are investigated in detail. The study is carried out in the perspective of pseudo-differential operators, quantization and signal analysis. Although the emphasis is put on locally compact and Hausdorff groups, results in the context of homogeneous spaces are given in order to unify the various localization operators into a single theory. Several new spectral results on pseudo-differential operators in the setting of localization operators are presented for the first time. The book is accessible to graduate students and mathematicians who have a basic knowledge of measure theory and functional analysis and wish to have a fast track to the frontier of research at the interface of pseudo-differential operators, quantization and signal analysis.

Table of Contents

Preface
Introduction	p. 1
Schatten-von Neumann Classes	p. 11
Topological Groups	p. 21
Haar Measures and Modular Functions	p. 25
Unitary Representations	p. 34
Square-Integrable Representations	p. 39
Wavelet Transforms	p. 48
A Sampling Theorem	p. 51
Wavelet Constants	p. 53
Adjoints	p. 57
Compact Groups	p. 60
Localization Operators	p. 63
S[subscript p] Norm Inequalities, 1 [less than or equal to] p [less than or equal to] [infinity]	p. 67
Trace Class Norm Inequalities	p. 71
Hilbert-Schmidt Localization Operators	p. 79
Two-Wavelet Theory	p. 84
The Weyl-Heisenberg Group	p. 90
The Affine Group	p. 98
Wavelet Multipliers	p. 107
The Landau-Pollak-Slepian Operator	p. 113
Products of Wavelet Multipliers	p. 117
Products of Daubechies Operators	p. 124
Gaussians	p. 129
Group Actions and Homogeneous Spaces	p. 141
A Unification	p. 143
The Affine Group Action on R	p. 147
References	p. 149
Index	p. 155
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