Operator Theory and Inverse Problems

University Of California-Santa Barbara, Santa Barbara CA

Investigators

Abstract

Operator Theory and Inverse Problems Abstract The proposed research will focus on a few inverse problems: the uniqueness in the multivariable moment problem, the best approximation of planar domains with finitely many prescribed moments; the inverse spectral problem for the modulus of the restriction operator between the Bergman spaces of two planar domains; a renormalized Riesz transform with applications to image reconstruction in any dimension. Methods of operator theory, function theory and approximation theory will be combined in this work. In many areas of modern activity it is required to reconstruct an entity from partial, and sometimes distant and indirect, data. Inverse problems address such specific questions. The proposed research will be concerned with a couple of inverse problems arising for instance in tomography or in geophysics. To be more 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.