Positivity, Inverse Problems, and Operator Theory

University Of California-Santa Barbara, Santa Barbara CA

Investigators

Abstract

The project is centered on the study of mathematical aspects of inverse problems, from the point of view of functional and complex mathematical analysis. Statistics, spectral analysis, X-ray crystallography, tomography offer models and classical examples of such questions. More specifically, most of the modern inverse problems are related to (power, Fourier, wavelet) moments of mass distributions. In their turn, moment problems form a well established field of research with a fascinating history of about two centuries. The project focuses on specific moment problems arising in shape and volume reconstruction from distant measurements (for instance of electric or gravitational fields). An important component of the project is the study of the best approximation and the error bounds related to reconstruction algorithms. The success of previous researches of the principal investigator on similar two dimensional questions guarantee positive results in his attempt to expand his horizon to three or more dimensions. He will collaborate with several experts in pure and applied mathematics, as well as with graduate students and young research assistants. Their combined efforts and results will be significant for present studies in functional analysis, function theory and numerical mathematics, as well as for some specific branches of applied mathematics and engineering.