Interpolation and Sampling for Generalized Bergman Spaces on finite Riemann surfaces

The goal of this paper is to establish sufficient conditions for a uniformly separated set on a finite Riemann surface to be interpolating or sampling for a generalized Bergman space of holomorphic functions on that surface. Let us fix an open Riemann surface X. Much of the geometry used in the statements and proofs of our results arises from potential theory on X. If X is hyperbolic, then X admits a Green’s function, while if X is parabolic, then X admits a so-called Evans Kernel. (See Section 2 for definitions.) After a normalization of the latter, both objects are unique, and we refer to either as the extremal fundamental solution E(z, ζ). Associated to this extremal fundamental solution is a distance ρ(z, ζ) = e, and we denote by De(z) the e-disk with respect to this distance. The geometry and potential theory we use in this paper is discussed in greater detail in Section 2 To a conformal metric g = e−ψ|dz|2 on X, a smooth function φ : X → R and a discrete subset Γ ⊂ X, uniformly separated with respect to the distance ρ above, we associated the following two Hilbert spaces: