Bounded analytic functions on unbounded covering surfaces (Q1257130)

Let \(G\) be an open Riemann surface of hyperbolic type and \(R\) an unbounded \(n\)-sheeted covering surface over \(G\). Denote by \(\vert \zeta_k\vert\) the set of projections of branch points and by \(n_k\) the order of branching over \(\zeta_k\). \textit{H. L. Selberg} [Comment. Math. Helv. 9, 104--108 (1937; Zbl 0015.30805)] proved that if \(G\) is the unit disk \(\vert z\vert <1\) and \(z_0\in G - \{\zeta_k\}\), then there is an \(f\) in the algebra \(H^\infty(R)\) such that \(f\) assumes distinct values at any two points over \(z_0\) iff \[ \sum n_kg_G(\zeta_k,z_0) < \infty, \] where \(g\) is the Green's function on \(G\) with pole at \(z_0\). This was extended to a bounded plane region \(G\) of finite connectivity by \textit{Y. Yamamura} [Sci. Rep. Tokyo Kyoiku Daigaku, Sect. A 10, 88--102 (1969; Zbl 0175.36302)]. The author extends these results to a larger class of surfaces that satisfy an exhaustion condition of H. Widom; in particular, it is shown that the conditions in Selberg's theorem are equivalent to the condition that \(H^\infty(R)\) separates points of \(R\).