| We show that on a closed subset E of an arbitrary Riemann surface, if each continuous function analytic on the interior of E can be uniformly approximated on E by meromophic functions, then this property must hold locally on E. Since locally, we are essentially in the complex plane, this allows us to use and generalize an important theorem on approximation by rational functions due to A. G. Vitushkin. This in turn gives a complete characterization of the sets of meromorphic uniform approximation on Riemann surfaces of finite genus, an important subclass of Riemann surfaces.; We obtain necessary and sufficient conditions on an open subset of a Riemann surface in order that every bounded analytic function on this open set be a pointwise limit of a bounded sequence of uniformly continuous analytic functions on the open set. We also prove a similar result about bounded pointwise approximation by meromorphic functions. This generalizes to Riemann surfaces results obtained in the complex plane by T. W. Gamelin and J. Garnett.; We completely characterize the real harmonic functions f with the property that every continuous function on a bounded and closed set of a Riemann surface can be approximated by functions in the algebra generated by the holomorphic functions and a real-valued continuous function f. We also study the Cauchy transform of a measure, peak sets and antisymmetric sets on Riemann surfaces. Those are important concepts giving crucial information regarding the class of holomorphic functions on Riemann surfaces. |
