RUI: Interpolation and Sampling in Bergman Spaces; Factors of Harmonic Polynomials

San Francisco State University, San Francisco CA

Investigators

Abstract

The methods used to characterize interpolation sequences vary depending upon whether the context is the Hardy space, the Bergman space, the Fock space, or the Dirichlet space. This research will attempt to unify these theories, using the concept of weak interpolation, which can be defined in the general context of a Hilbert space of analytic functions with a reproducing kernel. Moving away from the Hilbert space setting, an additional avenue of research will focus on obtaining a geometric characterization of interpolating sequences for the Bloch space. Yet another direction of research, suitable for experimental computer work by undergraduates who could search for useful evidence, involves the question of polynomial factors of harmonic polynomials in Euclidean spaces of arbitrary dimension. A polynomial that has an isolated zero at the origin cannot be a factor of a nonzero harmonic polynomial. This raises the question of which polynomials are factors of nonzero harmonic polynomials. Harmonic and analytic functions have been the subject of intense scrutiny by mathematicians, scientists, and engineers for the past two centuries. The Laplacian operator, whose vanishing characterizes harmonic functions, appears naturally in several areas of science and engineering, ranging from the heat equation to the distribution of electric charges. This project investigates properties of collections of analytic and harmonic functions in an attempt to understand deep aspects of the behavior of these functions. A key part of this project is the involvement of advanced undergraduates, who will be introduced to the world of mathematical research in the hope of inspiring them to pursue further studies and a career in mathematics. Showing these students the excitement of discovering new mathematical knowledge may help create the next generation of mathematicians.