Harmonic measures in approximation and orthogonal polynomials

University Of South Florida, Tampa FL

Investigators

Abstract

This is a three-year project in classical analysis that focuses, in particular, on approximation theory and orthogonal polynomials. A common unifying theme is the appearance of harmonic measures in various problems to be investigated, as well as the use of fast decreasing (Dirac-delta-like) polynomials to merge local approximants so as to create global ones. The applications include universality results in random matrix theory, various polynomial inequalities, and a domain reconstruction procedure based on Christoffel functions. Other parts of the project are related to nonclassical orthogonal polynomials/Christoffel functions, to polynomial approximation in several variables, and to developing new tools (based on fast decreasing polynomials) for their study. The intellectual merit lies in achieving a better understanding of the role of harmonic measures in approximation theory and orthogonal polynomials, as well as in the manifestation of the usefulness of fast decreasing polynomials in various areas of mathematics somewhat remote from the main topic of the project. For example, upon successful completion of the project the Principal Investigator will have a characterization (in terms of smoothness) of the rate of polynomial approximation on general convex domains, which will close a classical line of research going back more than half a century. The techniques to be applied will include systematic usage of harmonic analysis in approximation theory, thereby offering new tools in the latter field. The proposed research will have broader impact beyond approximation theory and orthogonal polynomials, for the results are relevant to other branches of mathematics, physics, and engineering. For example, various forms of the so-called universality conjecture have originated from studies in theoretical and statistical physics, studies that showed a certain universal local behavior that was independent of the original quantities describing the given system. In some form the mathematical formulation translates into a fine local behavior of orthogonal polynomials, and the proposal offers a tool to study this fine behavior and to prove rigorously universality under general conditions. Another area where the proposed research on orthogonal polynomials has broader impact is domain/shape reconstruction from scanned data (MRI-type scans). A recent method is based on area-generated orthogonal polynomials, but that method assumed that the domain to be reconstructed was without holes/cavities (information that may not be true or may not be known in advance). The present proposal lays the theoretical foundation for a modification of the method applying to domains that may have holes. By a relatively simple iteration, the algorithm will not only reconstruct the outer boundary of the domain (the original goal), but it will actually allow the reconstruction of the inner holes as well. Broader impacts of the research also include stimulating interest in undergraduates and mathematics majors and enhancing the research environment for them. Graduate and Ph.D. students will have the opportunity to learn the basic results and techniques of several different disciplines, as well as their interrelations. Special emphasis will be made in outreach to the general public. This will be achieved by publishing educational articles connected with the research, thereby offering understanding and appreciation for science. Relevant lectures will be held at different levels from undergraduate societies to professionals meetings.