Approximation by rational functions on Riemann surfaces

Abstract:

In this paper, we show that if $F \in {L^p}(k,\alpha )$ on $\Gamma$ where $\Gamma$ denotes the border of a compact bordered Riemann surface $\bar R$, then F can be uniquely written as the sum of a function in ${H^p}(k,\alpha )$ and a function in ${G^p}(k,\alpha )$ and moreover that F can be approximated on $\Gamma$ in ${L^p}$ norm to within $A/{n^{k + \alpha }}$ by a sequence of rational functions on the union of $\bar R$ with its double.