Harmonic and Applied Analysis

From Groups to Signals

  • Stephan Dahlke
  • Filippo De Mari
  • Philipp Grohs
  • Demetrio Labate

Part of the Applied and Numerical Harmonic Analysis book series (ANHA, volume 68)

Table of contents

  1. Front Matter
    Pages i-xv
  2. Stephan Dahlke, Filippo De Mari, Philipp Grohs, Demetrio Labate
    Pages 1-5
  3. Filippo De Mari, Ernesto De Vito
    Pages 7-81
  4. Stephan Dahlke, Sören Häuser, Gabriele Steidl, Gerd Teschke
    Pages 83-147
  5. Philipp Grohs
    Pages 199-248
  6. Back Matter
    Pages 249-256

About this book

Introduction

This contributed volume explores the connection between the theoretical aspects of harmonic analysis and the construction of advanced multiscale representations that have emerged in signal and image processing. It highlights some of the most promising mathematical developments in harmonic analysis in the last decade brought about by the interplay among different areas of abstract and applied mathematics. This intertwining of ideas is considered starting from the theory of unitary group representations and leading to the construction of very efficient schemes for the analysis of multidimensional data.

After an introductory chapter surveying the scientific significance of classical and more advanced multiscale methods, chapters cover such topics as

  • An overview of Lie theory focused on common applications in signal analysis, including the wavelet representation of the affine group, the Schrödinger representation of the Heisenberg group, and the metaplectic representation of the symplectic group
  • An introduction to coorbit theory and how it can be combined with the shearlet transform to establish shearlet coorbit spaces
  • Microlocal properties of the shearlet transform and its ability to provide a precise geometric characterization of edges and interface boundaries in images and other multidimensional data
  • Mathematical techniques to construct optimal data representations for a number of signal types, with a focus on the optimal approximation of functions governed by anisotropic singularities.

A unified notation is used across all of the chapters to ensure consistency of the mathematical material presented.

Harmonic and Applied Analysis: From Groups to Signals is aimed at graduate students and researchers in the areas of harmonic analysis and applied mathematics, as well as at other applied scientists interested in representations of multidimensional data. It can also be used as a textbook for

graduate courses in applied harmonic analysis.​

Keywords

Coorbit Spaces Fourier Analysis Group Representation Theory Harmonic Analysis Lie Theory Shearlets Signal Analysis Symplectic Group Unitary Representation

Editors and affiliations

  • Stephan Dahlke
    • 1
  • Filippo De Mari
    • 2
  • Philipp Grohs
    • 3
  • Demetrio Labate
    • 4
  1. 1.Mathematics and Computer SciencesPhilipps-Universität MarburgMarburgGermany
  2. 2.Department of MathematicsUniversità di GenovaGenovaItaly
  3. 3.Applied MathematicsETH ZürichZürichSwitzerland
  4. 4.Department of MathematicsUniversity of HoustonHoustonUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-319-18863-8
  • Copyright Information Springer International Publishing Switzerland 2015
  • Publisher Name Birkhäuser, Cham
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-319-18862-1
  • Online ISBN 978-3-319-18863-8
  • Series Print ISSN 2296-5009
  • Series Online ISSN 2296-5017
  • About this book