Mathematical Sciences: An Abstract Theory of Wavelets

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The purpose of this research project is to formulate in an abstract setting the notions and theory involved in the important area of wavelet bases. The research intends to derive results about the existence of wavelets, quality of wavelets, and summability results for wavelet expansions. In the abstract context, one may bring to bear the powerful and modern theories of unitary group representations, Von Neumann algebras, etc. Preliminary results indicate that this abstraction can lead to generalizations of and improvements of the classical results. Wavelets and multi-resolutions are proving to be suitable replacements for Fourier analysis in a variety of theoretic and applied problems, especially in signal processing. This project, in the general area of modern analysis, provides a wavelet description and theory within the context of unitary operators on the internal affine structure of a separable Hilbert space.