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Wandering Vectors for Unitary Systems and Orthogonal Wavelets
Xingde Dai, University of North Carolina, Charlotte, NC, and David R. Larson, Texas A&M, College Station, TX
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Memoirs of the American Mathematical Society
1998; 68 pp; softcover
Volume: 134
ISBN-10: 0-8218-0800-1
ISBN-13: 978-0-8218-0800-9
List Price: US$42
Individual Members: US$25.20
Institutional Members: US$33.60
Order Code: MEMO/134/640
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This volume concerns some general methods for the analysis of those orthonormal bases for a separable complex infinite dimensional Hilbert space which are generated by the action of a system of unitary transformations on a single vector, which is called a complete wandering vector for the system. The main examples are the orthonormal wavelet bases. Topological and structural properties of the set of all orthonormal dyadic wavelets are investigated in this way by viewing them as complete wandering vectors for an affiliated unitary system and then applying techniques of operator algebra and operator theory.

Features:

  • describes an operator-theoretic perspective on wavelet theory that is accessible to functional analysts
  • describes some natural generalizations of standard wavelet systems
  • contains numerous examples of computationally elementary wavelets
  • poses many open questions and directions for further research

This book is particularly accessible to operator theorists and operator algebraists who are interested in a functional analytic approach to some of the pure mathematics underlying wavelet theory.

Readership

Research mathematicians, engineers and graduate students interested in functional analysis and/or wavelet theory; computer scientsts.

Table of Contents

  • Introduction
  • The local commutant
  • Structural theorems
  • The wavelet system \(\langle D,T\rangle\)
  • Wavelet sets
  • Operator interpolation of wavelets
  • Concluding remarks
  • Appendix: Examples of interpolation maps
  • Bibliography
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