Interpolation and sampling for generalized Bergman spaces on finite Riemann surfaces. (English) Zbl 1167.30028

The aim of this paper is to demonstrate the relations between the potential theory of a Riemann surface and its interpolation and sampling properties. Sufficient conditions are given for a uniformly separated set on a finite Riemann surface to be interpolating or sampling for a generalized Bergmann space of holomorphic functions on the surface. A wide survey of the analytic geometry of Riemann surfaces is given and the fundamental metric is defined. Examples such as the Euclidean plane and the disk are given. After recalling the construction of finite Riemann surfaces, the authors deal with analytical geometric properties of these surfaces.


30F99 Riemann surfaces
30F15 Harmonic functions on Riemann surfaces
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