Mathematical Sciences: Complex Analysis and Harmonic Analysis

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Work will be directed at problems arising in the theories of several complex variables and harmonic analysis. The research represents a continuation of earlier investigations following three themes: convolution equations, interpolation problems and ideals generated by exponential polynomials. The background for problems of convolution and interpolation can be traced to fundamental questions of convolutions of a single function by a collection of measures vanishing simultaneously, and whether the system can be analyzed through the joint spectrum of the measures' Fourier transforms. Work on such questions led to conditions which, while correct, are not easily verified. The principal investigator will pursue several concrete-type criteria on which there has already been partial success. One particular impediment leads to a number-theoretic question which estimates the distance between roots of exponential polynomials. Some progress has been made here which may lead to applications to delay-type partial differential equations. A different type of convolution problem which appears in other contexts is that of the Pompeiu problem. In its rudimentary form, the problem concerns integrals of an unknown function taken over some geometric shape as the shape (or shapes) are moved by rigid motions. Whether or not one can recover the function from these calculations is a deep question whose resolution is known only in cases involving considerable symmetry. The principal investigator has expanded his work to the setting of manifolds (symmetric spaces). He has shown that the question of failure of the Pompeiu problem is directly related to a Neumann boundary value problem for an elliptic partial differential equation. While there are many applications of the result, a condition is imposed on the manifolds - they must be of rank 1. Work will be done in easing this requirement. This research has close connections to tomographic investigations.