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.
Abstract Harmonic Analysis of Continuous Wavelet Transforms
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Editorial Reviews
Review
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 Rösler, 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)
From the Back Cover
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 book can also be read as a problem-driven introduction to the Plancherel formula.
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Inside This Book
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First Sentence:
In this chapter we present the representation-theoretic approach to continuous wavelet transforms. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
convolution idempotents, quasiregular representation, direct integral view, measurable transversal, discrete series case, normalized tight frame, nonunimodular groups, conull set, admissible vectors, sampling subspace, coherent state system, unimodular case, central decomposition, commuting algebra, direct integral representation, transform side, discrete series representations, direct integral decomposition, measurable field, measure decomposition, dyadic wavelet transform, measurable structure, admissibility criterion, admissibility criteria, admissibility conditions Key Phrases - Capitalized Phrases (CAPs): (learn more)
Mackey Borel, Proof of Corollary New!
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Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
In this chapter we present the representation-theoretic approach to continuous wavelet transforms. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
convolution idempotents, quasiregular representation, direct integral view, measurable transversal, discrete series case, normalized tight frame, nonunimodular groups, conull set, admissible vectors, sampling subspace, coherent state system, unimodular case, central decomposition, commuting algebra, direct integral representation, transform side, discrete series representations, direct integral decomposition, measurable field, measure decomposition, dyadic wavelet transform, measurable structure, admissibility criterion, admissibility criteria, admissibility conditions Key Phrases - Capitalized Phrases (CAPs): (learn more)
Mackey Borel, Proof of Corollary New!
Books on Related Topics | Concordance | Text Stats Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
Citations (learn more)
This book cites 100
books:
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- Geometric Curve Evolution and Image Processing (Lecture Notes in Mathematics) by Frédéric Cao in Back Matter
- Heegner Modules and Elliptic Curves (Lecture Notes in Mathematics) by Martin L. Brown in Back Matter
- Big Queues (Lecture Notes in Mathematics) by Ayalvadi J. Ganesh in Back Matter
- Grothendieck Duality and Base Change (Lecture Notes in Mathematics) by Brian Conrad in Back Matter
- Hardy Spaces on Homogeneous Groups. (MN-28): (Mathematical Notes) by Gerald B. Folland in Back Matter
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