Mathematical Sciences: Wavelets Frames and Discrete Group Representations

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DMS-9500269 PI: Johnston Johnston will use the unitary representation theory of discrete groups to unify the continuous and discrete wavelet transform theories. The square-integrability of unitary representations of certain discretized countable subgroups of the real affine group and the real Heisenberg group will be investigated. This will give new frame constructions as well as group-theoretic proofs for well-known results about discrete wavelet sets in Hilbert spaces. Harmonic analysis combines those elements of mathematics best exemplifying the ideas of synthesis. One seeks to decompose complex problems into fundamental components. These components are then analyzed for their basic characteristics. Finally, the solution is reconstructed through a recombination of the components. The Fourier series and Fourier transform are examples of tools used in this context; one discrete , the other representing a continuous decomposition. More recently the theory of oscillatory integrals added new dimensions to some of the more classical approaches to harmonic analysis.