Mathematical Sciences: Wavelets: Theory and Application

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Daubechies 9401785 Work supported by this award focuses on the mathematical development of wavelet analysis and its applications. This is a branch of harmonic analysis closely allied with applications to many areas of science. The underlying goals of wavelet analysis are (i) to determine functions whose integer translates and dyadic dilates form orthogonal (or biorthogonal) bases for square-integrable functions (ii) to ensure that the basis elements localize both time and frequency and (iii) that they have the requisite smoothness dictated by the problem. Normally, wavelets are also expected to have compact support. In this project five goals are to be addressed. They include analysis of the smoothness of infinitely supported wavelets, the construction of redundant families of analytic wavelets associated with a multiresolution analysis and a study of the consequences of truncating wavelet filter coefficients and finding ways of doing the trunction in a stable manner. Work will also be done in applying wavelets, with new filters, and a new nonlinear squeezing technique, to speech analysis. Finally, effort will be made to construct a special family of multifractal functions for which the so-called thermodynamic formalism can be verified explicitly, and which can be used as a mathematical laboratory. The theory of wavelets, as it has become known, is actually a body of ideas which has developed dramatically over the past decade, following several important discoveries by physicists, computer scientists and engineers concerned with signal processing and data compression. It evolved into a synthesis of many existing techniques into a framework which offers possibilities for improved applications and challenging mathematical ideas which will require years to reach what might be considered a mature stage. Many areas of science are now adapting wavelet constructs to important problems currently under investigation. These included statisticians lo oking for patterns in large data sets, compression of fingerprint data, edge reconstruction of images and the generation of fractal sets with prescribed characteristics.