FRG: Collaborative Research: Focused Research on Wavelets, Frames, and Operator Theory

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FRG Collaborative ProposalPROPOSAL NUMBER PI INSTITUTION0139759 Benedetto University of Maryland, lead PI0139740 Aldroubi Vanderbilt University0139473 Jorgensen University of Iowa0139261 Heil, Wang Georgia Institute of Technology0139366 Baggett Univ. of Colorado0139783 Olafsson Lousiana State Univ.0139386 Larson Texas A&MABSTRACT:Fundamental problems are addressed in wavelet theory, non-uniformsampling, frames, and the theory of spectral-tile duality. Theseproblems are inextricably interwoven by concept andtechnique. Operator theory provides the major unifying framework,combined with an integration of ideas from a diverse spectrum ofmathematics including classical Fourier analysis, noncommutativeharmonic analysis, representation theory, operator algebras,approximation theory, and signal processing. For example, theconstruction, implementation, and ensuing theory of single dyadicorthonormal wavelets in Euclidean space requires significant inputfrom all of these disciplines as well as deep spectral-tile results.There is intrinsic mathematical importance in the aforementioned problems, and the solutions to be formulated have broad and creative implications, both for mathematics and for applications in engineering and physics. The topics of this project have direct bearing on fast acquisition and motion problems in MRI, as well as in formulating algorithms for compression and noise reduction by means of proper cochlear modelling. There are furtherapplications in quantum computing and image processing, and the development of non-uniform sampling strategies by this project play a role in state of the art A/D conversion methods used in multifunction RF systems. These interdisciplinary applications depending on modern mathematical analysis have educational implications in terms of cross-fertilization of ideas and researchopportunities for graduate students.