Mathematical Sciences: Complex Analysis and Harmonic Analysis

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This project continues mathematical research focusing on complex analytic methods applied to the analysis of convolution equations, integral geometry and systems of algebraic equations. Earlier work on interpolation problems in spaces of entire functions has led naturally to questions concerning multidimensional convolutions equations. One is actually interested in the question of deconvolution. Many physical realizable systems such as transducers and sensors can be modeled as convolution with distributions. This gives rise to a scheme to estimate signals such as images, radiation, acoustics, etc. The data are given as a convolution plus noise and the reconstruction process in this project consists of finding systematic ways of constructing deconvolvers. In pursuing this work, several mathematical problems arise which are treated as distinct objectives. They include finding the degrees and height in the algebraic Bezout equation, obtaining new properties and representations of residue currents (multidimensional residues) and questions in integral geometry related to the Pompieu problem. The Pompieu problem actually covers a vast array of questions concerning integrals of a fixed function over various congruent configurations. If the integtrals equal zero on a class of congruent sets, one is interested in knowing whether or not the function must be zero. Most of this work is set within the framework of several complex variables. Techniques of analytic function theory and harmonic analysis are predominant. Diverse applications to systems theory and engineering are expected.