Generalized Multiresolution Analyses

1. Front Matter

   Pages i-x

   PDF

2. Introduction

   • Kathy D. Merrill

   Pages 1-8

3. The Invariance of the Core Subspace

   • Kathy D. Merrill

   Pages 9-21

4. The Multiplicity Function

   • Kathy D. Merrill

   Pages 23-36

5. Wavelet Sets

   • Kathy D. Merrill

   Pages 37-54

6. Generalized Filters

   • Kathy D. Merrill

   Pages 55-74

7. Fractal Spaces

   • Kathy D. Merrill

   Pages 75-83

8. Composite Dilations and Crystallographic Groups

   • Kathy D. Merrill

   Pages 85-96

9. Abstract Constructions of GMRAs

   • Kathy D. Merrill

   Pages 97-110

10. Back Matter

    Pages 111-113

    PDF

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.

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

“This book will be appreciated mainly by graduate students and researchers, who have been already introduced to classical wavelet theory, and have interest in the development of new construction methods of wavelets, and their use in more general contexts.” (Sandra Saliani, Mathematical Reviews, December, 2019)

“It is well-suited for graduate students who want to become familiar with this interesting topic and to researchers who would like to get a quick overview over the area.” (Peter Massopust, zbMATH 1409.42001, 2019)

• Department of Mathematics, The Colorado College, Colorado Springs, USA

  Kathy D. Merrill

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