Ebook: Wavelet Transforms and Localization Operators

This book is based on lectures given at the Global Analysis Research Center (GARC) of Seoul National University in 1999and at Peking University in 1999and 2000. Preliminary versions of the book have been used for various topics courses in analysis for graduate students at York University. We study in this book wavelet transforms and localization operators in the context of infinite-dimensional and square-integrable representations of locally compact and Hausdorffgroups. The wavelet transforms studied in this book, which include the ones that come from the Weyl-Heisenberg group and the well-known affine group, are the building blocks of localization operators. The theme that dominates the book is the spectral theory of wavelet transforms and localization operators in the form of Schatten-von Neumann norm inequalities. Several chap­ ters are also devoted to the product formulas for concrete localization operators such as Daubechies operators and wavelet multipliers. This book is a natural sequel to the book on pseudo-differential operators [103] and the book on Weyl transforms [102] by the author. Indeed, localization operators on the Weyl-Heisenberg group are Weyl transforms, which are in fact pseudo-differential operators. Details on the perspective and the organization of the book are laid out in the first chapter. This is a book on mathematics and is written for anyone who has taken basic graduate courses in measure theory and functional analysis. Some knowledge of group theory and general topology at the undergraduate level is also assumed.

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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.

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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.
Content:
Front Matter....Pages i-vii
Introduction....Pages 1-10
Schatten-von Neumann Classes....Pages 11-20
Topological Groups....Pages 21-24
Haar Measures and Modular Functions....Pages 25-33
Unitary Representations....Pages 34-38
Square-Integrable Representations....Pages 39-47
Wavelet Transforms....Pages 48-50
A Sampling Theorem....Pages 51-52
Wavelet Constants....Pages 53-56
Adjoints....Pages 57-59
Compact Groups....Pages 60-62
Localization Operators....Pages 63-66
Trace Class Norm Inequalities....Pages 67-70
Hilbert-Schmidt Localization Operators....Pages 71-78
Two-Wavelet Theory....Pages 79-83
The Weyl-Heisenberg Group....Pages 84-89
The Affine Group....Pages 90-97
Wavelet Multipliers....Pages 98-106
The Landau-Pollak-Slepian Operator....Pages 107-112
Products of Wavelet Multipliers....Pages 113-116
Products of Daubechies Operators....Pages 117-123
Gaussians....Pages 124-128
Group Actions and Homogeneous Spaces....Pages 129-140
A Unification....Pages 141-142
The Affine Group Action on ?....Pages 143-146
Back Matter....Pages 147-148
....Pages 149-156