Gaussian Polynomial Filters and Generalized Shift-Invariant Frames

We present and study a family of filters on $L^2(\mathbb{R}^d)$ consisting of Gaussian polynomials. That is, multipliers in the frequency domain that are products of polynomials and Gaussians. These filters are constructed to approximate the characteristic functions of fairly general sets in $\mathbb{R}^d$, in an almost-uniform sense. We also study generalized shift-invariant (GSI) frames for $L^2(\mathbb{R}^d)$. These are frames consisting of regular lattice translations of countably many functions, which we call generators. GSI frames are fundamental to sampling theory and many areas of applied mathematics and engineering, in particular, signal and image analysis. Their distinguishing feature is an accommodation for generators which may be un- related to one another, and for general lattices of translations, which may vary with the generators. GSI frames generalize systems such as wavelets, Gabor systems, shearlets, curvelets, filter banks, etc. We discuss very general conditions on the generators under which one can determine lattice spacings, or sampling rates, so as to meet the frame condition. We develop a fast and numerically stable method for inverting the frame operator, and we give a detailed analysis of this method, as well as of the fast numerical implementation of the synthesis and analysis operators associated with GSI frames. We give a careful analysis of two methods for obtaining approximate dual GSI frames for general GSI frames. We apply this GSI system framework to the the Gaussian polynomial filters developed in this dissertation to obtain frames of translated Gaussian polynomials.Mathematics, Department o