Ein Approximationssatz für holomorphe Funktionen auf nicht-kompakten Riemannschen Flächen

Abstract

Let R be a non-compact Riemann surface andO(R) the algebra of all holomorphic functions on R. A subalgebraA ⊂O(R) is calledfull (“voll”), if (F1) for every point ϕɛR there is a function f∈A with a simple zero at ϕ and no other zeros; (F2) if f, g∈A and f/g has no poles, then f/∈A. In 1971 Ian RICHARDS set the problem whether full subalgebras are dense inO(R), with respect to the topology of compact convergence. We answer this question in the positive, using a lemma of I. RICHARDS and theorems of R. ARENS and the author.

Does this approximation theorem remain true for Stein manifolds of dimension n>1, if one modifies condition (F1) in a natural way? A counterexample is provided by a domain of holomorphy G⊂⊄² and a full, but not dense subalgebraA ⊂O(G).