This book grew out of the
Special Session on
Wavelets, Frames, and Operator
that we organized at
Annual Meeting of the AMS in Baltimore,
15-18, 2003,
and an
immediately following NSF-sponsored workshop orga-
nized by John Benedetto at The University of Maryland,
19-21, 2003.
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
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.
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
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
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.
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