Weyl-Heisenberg Wavelet Expansions: Existence and Stability in Weighted Spaces

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## Abstract

The theory of wavelets can be used to obtain expansions of vectors in certain spaces. These expansions are like Fourier series in that each vector can be written in terms of a fixed collection of vectors in the Banach space and the coefficients satisfy a "Plancherel Theorem" with respect to some sequence space. In Weyl-Heisenberg expansions, the expansion vectors (wavelets) are translates and modulates of a single vector (the analyzing vector) . The thesis addresses the problem of the existence and stability of Weyl-Heisenberg expansions in the space of functions square-integrable with respect to the measure w(x) dx for a certain class of weights w. While the question of the existence of such expansions is contained in more general theories, the techniques used here enable one to obtain more general and explicit results. In Chapter 1, the class of weights of interest is defined and properties of these weights proven. In Chapter 2, it is shown that Weyl-Heisenberg expansions exist if the analyzing vector is locally bounded and satisfies a certain global decay condition. In Chapter 3, it is shown that these expansions persist if the translations and modulations are not taken at regular intervals but are perturbed by a small amount. Also, the expansions are stable if the analyzing vector is perturbed. It is also shown here that under more general assumptions, expansions exist if translations and modulations are taken at any sufficiently dense lattice of points. Like orthonormal bases, the coefficients in Weyl-Heisenberg expansions can be formed by the inner product of the vector being expanded with a collection of wavelets generated by a transformed version of the analyzing vector. In Chapter 4, it is shown that this transformation preserves certain decay and smoothness conditions and a formula for this transformation is given. In Chapter 5, results on Weyl-Heisenberg expansions in the space of square-integrable functions are presented.