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© 2005

Abstract Harmonic Analysis of Continuous Wavelet Transforms

  • Authors
Book

Part of the Lecture Notes in Mathematics book series (LNM, volume 1863)

Table of contents

  1. Front Matter
    Pages N2-X
  2. Hartmut Führ
    Pages 1-13
  3. Hartmut Führ
    Pages 139-168
  4. Hartmut Führ
    Pages 169-184
  5. Hartmut Führ
    Pages 185-190
  6. Back Matter
    Pages 191-199

About this book

Introduction

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.

Keywords

Heisenberg group Plancherel theory continuous wavelet transforms harmonic analysis sampling theory

Bibliographic information

  • Book Title Abstract Harmonic Analysis of Continuous Wavelet Transforms
  • Authors Hartmut Führ
  • Series Title Lecture Notes in Mathematics
  • Series Abbreviated Title LNM
  • DOI https://doi.org/10.1007/b104912
  • Copyright Information Springer-Verlag Berlin Heidelberg 2005
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
  • Softcover ISBN 978-3-540-24259-8
  • eBook ISBN 978-3-540-31552-0
  • Series ISSN 0075-8434
  • Series E-ISSN 1617-9692
  • Edition Number 1
  • Number of Pages X, 193
  • Number of Illustrations 0 b/w illustrations, 0 illustrations in colour
  • Topics Abstract Harmonic Analysis
    Fourier Analysis
  • Buy this book on publisher's site

Reviews

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)