Interpolation and sampling for generalized Bergman spaces on finite Riemann surfaces

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- Cerdà, 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. Unlike previous works, we introduce a family of densities all of which provide sufficient conditions. Thus we obtain new results even in classical cases, some of which might be useful in industrial applications. The main point of the article is to demonstrate the interaction between the potential theory of the Riemann surface and its interpolation and sampling properties.