Interpolation and Sampling on Riemann Surfaces

Abstract

We find sufficient conditions for a discrete sequence to be interpolating or sampling for certain generalized Bergman spaces on open Riemann surfaces. As in previous work of Bendtsson, Ortega-Cerda, Seip, Wallsten and others, our conditions for interpolation and sampling are as follows: If a certain upper density of the sequence has value less that 1, then the sequence is interpolating, while if a certain lower density has value greater than 1, then the sequence is sampling. However, unlike previous work, we construct an infinite number of densities, naturally parameterized by $L^1_{loc} ([0,\infty))$. These densities provide (different) sufficient conditions for interpolation and sampling. As a consequence of this flexibility, we obtain new results even in the classical cases of the Bargmann-Fock space (of entire functions integrable with respect to a Gaussian weight) and the classical Bergman space of L^2 holomorphic functions on the unit disk.