Abstract harmonic analysis of continuous wavelet transforms.(English)Zbl 1060.43002

Lecture Notes in Mathematics 1863. Berlin: Springer (ISBN 3-540-24259-7). x, 193 p. (2005).
It is well known that wavelets and Gabor systems can be viewed from a group-theoretical point of view: both systems appear by a unitary group representation of a locally compact group acting on $$L^2(R)$$. This book deals with generalizations, valid for general locally compact groups and unitary representations on arbitrary Hilbert spaces. The book is based on research papers by Führ and his coauthors. 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. The book contains six chapters. Chapter 1 introduces the continuous wavelet transform and the inversion formula; it further gives an overview of the book, and a description of some of the basic tools like group representations, operator algebras and Hilbert–Schmidt operators. Chapter 2 begins with an introduction to the Haar measure and Weil’s integral formula; after that, the coherent states and the coherent state expansions are introduced (the latter are also known as expansions via tight continuous frames). A detailed discussion of the admissibility condition is given. Chapter 3 deals with the Plancherel transform for locally compact groups. It contains a description of direct integrals and Mackey theory. Chapter 4 concerns the inversion of the Plancherel transform. Chapter 5 presents explicit versions of the admissibility criteria for group extensions. It also introduces the Weyl-Heisenberg systems (Gabor systems) and the Zak transform. The Weyl-Heisenberg system appears via the action of the Schrödinger representation on the Heisenberg group; the Heisenberg group is also the topic for Chapter 6, which characterizes the sampling spaces using the Plancherel transform.

MSC:

 43A30 Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc. 42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems 43A80 Analysis on other specific Lie groups
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