n-th Root Optimal Rational Approximants to Functions with Polar Singular Set

Abstract

Let be a bounded Jordan domain and be its complement on the Riemann sphere. We investigate the -th root asymptotic behavior in of best rational approximants, in the uniform norm on , to functions holomorphic on having a multi-valued continuation to quasi every point of with finitely many branches. More precisely, we study weak convergence of the normalized counting measures of the poles of such approximants as well as their convergence in capacity. We place best rational approximants into a larger class of -th root optimal meromorphic approximants, whose behavior we investigate using potential-theory on certain compact bordered Riemann surfaces.