RUI: Sampling and Interpolation on Riemann Surfaces and in Several Complex Variables

San Francisco State University, San Francisco CA

Investigators

Abstract

ABSTRACT Sampling and interpolation sequences have been characterized for many classical Hilbert spaces of analytic functions, such as the Hardy, Bergman, Fock, Dirichlet and Paley-Wiener space, but in all of these cases the domains of analyticity of the functions under consideration are either the disk or plane. The PI has obtained sufficient conditions for sampling and interpolation on Hilbert spaces of functions that are analytic on a large class of Riemann surfaces, but many questions are left open in that paper. Some are directly related to the question of sampling and interpolation, but others deal more generally with certain ideas introduced in the article and how they are related to classical invariants of open Riemann surfaces. Very little is known about sampling and interpolation sequences on domains in several complex variables, but the PI, in collaboration with others, has obtained sufficient conditions for hypersurfaces to be sampling or interpolating. While this does lead to some new conditions on sequences, the authors are hopeful that the ideas can be applied to obtain a complete characterization. Sampling and interpolation are of fundamental importance in mathematics and in the sciences at large. The particular spaces of functions mentioned above are important in applications for many reasons, the most significant of which is that they consist of functions with ``finite energy''. The problem of the interpolation of functions has been a central theme since the early days of complex analysis, while the sampling problem is considerably more recent. Sampling was popularized by Shannon in the 1940s as a significant portion of his theory of information. Since that time, engineers have been deeply interested in matters of sampling, so much so that the notion of sampling is now in the collective consciousness of everyday society. In a nutshell, sampling a function consists of measuring the value of a function at a certain sufficiently large but discrete set of points. The measurements should be made often enough so that the function is completely determined by its detected values. The interpolation problem can then be seen as the dual problem of reconstructing the function from this discrete, or ``digital'' data. An important aspect of this proposal involves undergraduate student projects dealing with open problems in Complex Analysis. One such problem is the question of which polynomials (in several real variables) are factors of harmonic polynomials. Another involves the determination of the best constant in the Bergman space maximum principle. Both of these questions are amenable to experimental work by students to gather evidence for an appropriate conjectured solution.