Harmonic measures, potentials, approximation and polynomials inequalities on general sets

University Of South Florida, Tampa FL

Investigators

Abstract

Research will be done in connection with metric properties of harmonic measures on infinitely connected domains, in connection with polynomial inequalities and orthogonal polynomials. The estimates for harmonic measures/Green functions will be given via a local integral involving a density function associated with the boundary of the domain. Such results directly lead to estimates for polynomials in the complex plane and to extensions of some classical polynomial inequalities for general compact sets. The results, in particular, will allow to find the order of the Markoff factors for Cantor-type sets, which in turn will produce compact sets of zero measure for which the Markoff factors are of the minimal growth order O(n^2). Weighted polynomial inequalities with respect to general (say doubling) weights will also be studied together with applications in connection with approximation theory and orthogonal polynomials. Another part of the research is polynomial approximation and orthogonal polynomials with varying weights (the weight varies with the degree), which will be studied with the help of logarithmic potentials in the presence of an external field. The approximation problem is fundamental in proving appropriate asymptotics for Christoffel functions and orthogonal polynomials with varying weights, that in turn can be used to test the universality hypothesis in statistical- mechanical models of quantum physics. The proposed research uses different tools from classical analysis, but the main method is potential thoretic. Research will be done in mathematical analysis. The main effort will be to find new ways to estimate some classical quantities and to extend some well known results to more general situations. These will allow wider applicability of several classical tools and will provide some new insights into the theoretical aspects of some fundamental questions in approximation theory and orthogonal polynomials.