Operator Algebras and Wavelet Theory

Texas A&M Research Foundation, College Station TX

Investigators

Abstract

ABSTRACT: The plan of this proposal is to utilize operator-algebraic methods to solve several problems in the mathematical theory of wavelets, and in the related area of frame theory, including Weyl-Heisenberg and Gabor theory. We have a new lead on the well-known wavelet connectivity problem, which we perceive to be a basic issue in the subject. Another wavelet problem of a basic issue nature concerns the question of when a Riesz wavelet which is known to be a linear combination of MRA wavelets is itself an MRA wavelet. Others problems we propose to work on concern norm-density of the wavelet frames, operator-theoretic interpolation of wavelets, superframes and superwavelets, and Weyl-Heisenberg frames. We will also work on two projects in a different direction concerning operator spaces, reflexivity and optimization. The first concerns an axiomatic description of a ranking-function for an abstract operator space. There is a natural definition of a concrete ranking-function for a concretely given operator space, but an abstract description of this is elusive and seems to be a basic issue. The second concerns the question of which matrix completion problems are well-posed in the sense of optimization theory. This proposal represents work of an interdisciplinary nature on the mathematics of wavelet and frame theory. Work in this direction that was previously supported by NSF has settled some open questions and has impacted the work of others in harmonic analysis and applications-oriented wavelet theory. Continuing in this direction is the main thrust of the present proposal. Numerous papers have been written in the past dozen years dealing with applications of wavelets to signal and image processing. So far, most of the published work has dealt directly with applications, and relatively little has been accomplished concerning the basic mathematical underpinnings of the subject. This proposal is concerned with this mathematics. There have been some surprises that have come up in our work in the past two years, and these discoveries have led to some potential areas of applications of a previously unsuspected nature. There are several outstanding problems we emphasize in this proposal and plan to work on. Under prior NSF support we also accomplished research with several co-authors on some basic problems in operator theory, operator spaces, operator algebras and matrix optimization, and this leads to some further open problems and directions we plan to pursue. Several graduate and undergraduate students arinvolved in this project. This grant also contains the NSF partial support of the annual Great Plains Operator Theory Symposium, a major mathematics research conference which rotates among a number of universities in the USA.