Preface

This book grew out of the

Special Session on

Wavelets, Frames, and Operator

Theory

that we organized at

the

2003

Annual Meeting of the AMS in Baltimore,

January

15-18, 2003,

and an

immediately following NSF-sponsored workshop orga-

nized by John Benedetto at The University of Maryland,

January

19-21, 2003.

Both

events were associated with the NSF Focused Research

Group (FRG) of which we

are a part, and whose other members are Akram Aldroubi,

Lawrence W. Baggett,

John J. Benedetto, Gestur Olafsson, and Yang Wang. The speakers in the Special

Session and the Maryland workshop were invited to contribute papers, and this

volume is the very pleasant result.

We hope that those

events and more like them that have since taken place

or are planned for the future, and the present book

itself, will act as a catalyst,

encouraging members of our community to work on one

or more of the many facets

of the intertwined subjects of wavelets, frames, and

operator theory. Some of

the papers included here focus more on one of the

three areas than the other

two, but all hint at exciting connections and

interrelationships. They stand at the

crossroads of harmonic analysis, operator theory, and applied mathematics. Some

papers are theoretical, some applied, but most are a mix

of theory and applications,

each inspiring the other. Wavelets and frames have emerged as significant tools in

mathematics and in technology over the past two decades. They interact with

harmonic analysis, with operator theory, and with a host of applications. In

their

primitive form, both wavelets and frames originate with the vector space notion of a

basis. They are used in the analysis of functions, and the functions make up infinite-

dimensional spaces, typically Hilbert spaces. While many wavelet constructions

yield orthonormal bases, frames by their very nature, including many interesting

classes of wavelets, satisfy conditions which are more general than the familiar

orthogonality relations. Historically, operator theory has played a big part in the

analysis of both wavelets and frames, and we hope to highlight this feature in our

collection of papers.

The workshops,

the research, and the publication of this volume were supported

in part by our FRG grant from the National Science Foundation.

1

It is also a

pleasure to thank Brian Treadway, whose assistance was essential to the smooth

1DMS-0139759 Collaborative Research: Focused Research on Wavelets, Frames, and Opera-

tor Theory. Description: In this project, fundamental problems are addressed in wavelet

theory,

non-uniform sampling, frames, and the theory of spectral-tile duality. These problems are in-

extricably interwoven by concept and technique. Operator theory provides the

major unifying

framework, combined with an integration of ideas from a diverse spectrum of mathematics

includ-

ing classical Fourier analysis, noncommutative harmonic analysis,

representation theory, operator

algebras, approximation theory, and signal processing. For example, the

construction, imple-

mentation, and ensuing theory of single dyadic orthonormal

wavelets in Euclidean space requires

significant input from all of these

disciplines as well as deep spectral-tile results.

vii