Convergence of two-point Pad\'e approximants to piecewise holomorphic functions

Abstract

Let $ f_0 $ and $ f_\infty $ be formal power series at the origin and infinity, and $ P_n/Q_n $, with $ \mathrm{deg}(P_n),\mathrm{deg}(Q_n)\leq n $, be a rational function that simultaneously interpolates $ f_0 $ at the origin with order $ n $ and $ f_\infty $ at infinity with order $ n+1 $. When germs $ f_0,f_\infty $ 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 $ might or might not separate the plane. We study uniform convergence of the approximants for the geometrically simplest sets $ F $ that do separate the plane.