Approximation by harmonic functions of closed subsets of Riemann surfaces (Q1119775)

This paper discusses various cases of the following general question: Given a closed set F on a Riemann surface \(\Omega\) and two continuous functions u, \(\epsilon\) on F with \(\epsilon (z)>0\) everywhere, find a harmonic function h on \(\Omega\) such that \(| h-u| <\epsilon\) on F. Both uniform (\(\epsilon\) constant) and better-than-uniform or Carleman approximation (any \(\epsilon (z)>0)\) is considered and the authors also consider the case where the approximating harmonic functions are allowed to have logarithmic singularities. One of their results is a geometric characterization of the sets F with the property that any continuous function on F which is harmonic in \(F^ 0\) can be approximated in the Carleman sense by functions harmonic in all of \(\Omega\).