Convergence of two-point Padé approximants to piecewise holomorphic functions

Abstract

Let and be formal power series at the origin and infinity, and , , be the rational function that simultaneously interpolates at the origin with order and at infinity with order . When germs and represent multi-valued functions with finitely many branch points, it was shown by Buslaev that there exists a unique compact set in the complement of which the approximants converge in capacity to the approximated functions. The set may or may not separate the plane. We study uniform convergence of the approximants for the geometrically simplest sets that do separate the plane. Bibliography: 26 titles.