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-Cerdá,
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.