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Generalized Multiresolution Analyses

  • Kathy D. Merrill

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

Also part of the Lecture Notes in Applied and Numerical Harmonic Analysis book sub series (LN-ANHA)

Table of contents

  1. Front Matter
    Pages i-x
  2. Kathy D. Merrill
    Pages 1-8
  3. Kathy D. Merrill
    Pages 9-21
  4. Kathy D. Merrill
    Pages 23-36
  5. Kathy D. Merrill
    Pages 37-54
  6. Kathy D. Merrill
    Pages 55-74
  7. Kathy D. Merrill
    Pages 75-83
  8. Kathy D. Merrill
    Pages 97-110
  9. Back Matter
    Pages 111-113

About this book

Introduction

This monograph presents the first unified exposition of generalized multiresolution analyses. Expanding on the author’s pioneering work in the field, these lecture notes provide the tools and framework for using GMRAs to extend results from classical wavelet analysis to a more general setting. 

Beginning with the basic properties of GMRAs, the book goes on to explore the multiplicity and dimension functions of GMRA, wavelet sets, and generalized filters. The author’s constructions of wavelet sets feature prominently, with figures to illustrate their remarkably simple geometric form. The last three chapters exhibit extensions of wavelet theory and GMRAs to other settings. These include fractal spaces, wavelets with composite dilations, and abstract constructions of GMRAs beyond the usual setting of L2(ℝn).

This account of recent developments in wavelet theory will appeal to researchers and graduate students with an interest in multiscale analysis from a pure or applied perspective. Familiarity with harmonic analysis and operator theory will be helpful to the reader, though the only prerequisite is graduate level experience with real and functional analysis.

Keywords

Generalized Multiresolution Analysis Wavelets Frames Filters Wavelet Sets Crystallographic groups Fractal spaces Multiscale Resolution

Authors and affiliations

  • Kathy D. Merrill
    • 1
  1. 1.Department of MathematicsThe Colorado CollegeColorado SpringsUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-319-99175-7
  • Copyright Information Springer Nature Switzerland AG 2018
  • Publisher Name Birkhäuser, Cham
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-319-99174-0
  • Online ISBN 978-3-319-99175-7
  • Series Print ISSN 2296-5009
  • Series Online ISSN 2296-5017
  • Buy this book on publisher's site