Harmonic measures, polynomial inequalities, orthogonal polynomials and approximation

University Of South Florida, Tampa FL

Investigators

Abstract

This is a program in classical mathematical analysis, in particular in potential theory, approximation theory and orthogonal polynomials. Some of the questions to be considered are fundamental in classical harmonic analysis, since they deal with smoothness properties of Green's functions and harmonic measures, which are basic objects both in the theory and its applications. The main problem is how smoothness of harmonic measures around boundary points is related to the geometry of the underlying domain. The answer to this problem is directly linked to problems in approximation theory, polynomial inequalities and estimates on orthogonal polynomials. The other direction of research is non-classical orthogonal polynomials, for which many questions are motivated by applications, and for which classical tools are not available. In particular, Dr. Totik will study orthogonal polynomials with respect to doubling measures which will open a new direction of research that seems to be the right one when the classical theory (as well as the related interpolation theory) is extended to more general classes. The results will have direct applications to interpolation, quadrature, numerical analysis and integral equations, and so are also relevant to physics and engineering. Harmonic analysis is a fundamental tool in many branches of science and engineering like signal processing, medical imaging, airplane design, pattern recognition, data compression etc. Approximation theory provides means to simplify theoretical models via replacing complicated functions/objects by simpler ones. The theory of orthogonal polynomials is extremely rich with applications in a vast array of problems like spectral theory, numerical analysis, electrostatics, statistical quantum mechanics, random matrices, birth and death processes, prediction theory, Radon transform and computer tomography, etc. The research will enhance our understanding of some areas of these field, as well as their interaction.