Operator Theory and Inverse Problems

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Operator Theory and Inverse Problems Abstract The proposed research will focus on a few inverse problems: the uniqueness in the multivariable moment problem, thebest approximation of planar domains with finitely many prescribedmoments; the inverse spectral problem for the modulus of therestriction operator between the Bergman spaces of two planardomains; a renormalized Riesz transform with applications toimage reconstruction in any dimension. Methods of operatortheory, function theory and approximation theory will be combinedin this work. In many areas of modern activity it is required to reconstructan entity from partial, and sometimes distant and indirect, data.Inverse problems address such specific questions. The proposedresearch will be concerned with a couple of inverse problemsarising for instance in tomography or in geophysics. To bemore specific, given a number of different angle X-rays of a body,we will propose a constructive, optimal way of approximating, or sometimes reconstructing this body.