Mathematical Sciences: Adaptive Frame Decompositions

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9307655 Gilbert This project will develop adaptive tools for decomposing function spaces that extend methods from Fourier analysis and wavelet theory. Specific functions are to be analyzed through the selection of a particular wavelet well-adapted to the function. Criteria for choosing the matched wavelet come from the specific application which creates the function. Families of analyzing wavelets will be designed having specific smoothness and decay properties needed for establishing such optimization criteria. Frame decompositions will also be studied in vector-valued settings in relation to extensions of the affine group and corresponding representation theory. This work will draw on a theory, developed recently which relates the generalized Cauchy- Riemann systems to unitarizable representations of groups containing the affine group. A particularly relevant example, previously studied by Torresani, is the semidirect product of the affine group with the Heisenberg group. Corresponding frame decompositions will incorporate aspects of both the short-time Fourier transform and wavelet expansions. Mathematical research focusing on wavelets and frame decompositions has its roots in classical harmonic analysis but derives its recent impetus from studies in signal processing. Here, the wavelet concept has proved remarkably effective in analyzing signals composed of both stationary and transient features. Traditional Fourier methods have not been effective tools in these studies. Applications to research on acoustics, data compression and partial differential equations has already proven the worth of the newly developing ideas. ***