Mathematical Sciences: An Abstract Theory of Wavelets and Multi-Resolution Analyses

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9401180 Baggett The purpose of this project is to continue formulating an abstract setting the notions and theorems involved with the important area of wavelet bases and multi-resolution analyses. Part of the project is to derive results about the existence of wavelets and MRA's, quality of the wavelets and MRA's, and summability results for wavelet expansions. In the abstract context, one may bring to bear the powerful and modern theories of unitary representations, von Neumann algebras, etc. Another part of the project is to use abstract theory to describe the notion of windowed Fourier transforms, which notion can also be formulated in terms of unitary representation theory and non-commutative harmonic analysis. Research on Wavelet Theory is an incredibly active area of mathematics. This is due mainly to the powerful applications to important areas involving high performance computing such as data compression and pattern recognition. Usually wavelet theory is approached using the real variable techniques of harmonic analysis. This project is aimed at formulating Wavelet Theory in an abstract mathematical setting. Powerful abstract mathematics can then be applied to get automatic existence and convergence theorems about wavelets. New insights into the fundamental nature of wavelets will occur. In a reciprocal manner, the salient features of wavelets will be reflected to produce new concepts in abstract mathematics. ***