Convergence of two-point Padé approximants to piecewise holomorphic functions

Abstract

Let f0 and f∞ be formal power series at the origin and infinity, and Pn/Qn, deg⁡(Pn),deg⁡(Qn)≤n, be the rational function that simultaneously interpolates f0 at the origin with order n and f∞ at infinity with order n+1. When germs f0 and f∞ represent multi-valued functions with finitely many branch points, it was shown by Buslaev that there exists a unique compact set F in the complement of which the approximants converge in capacity to the approximated functions. The set F may or may not separate the plane. We study uniform convergence of the approximants for the geometrically simplest sets F that do separate the plane.