On the Continuity of the Padé Operator

Abstract

Let $r_{m,n} $ be the Pade approximant of order $(m,n)$ for a given power series f. Let $T_{m,n} $, be the operator that maps f on $r_{m,n} $. It is known that $T_{m,n} $ satisfies a (local) Lipschitz condition in case $r_{m,n} $ is normal, but may fail to be continuous if $r_{m,n} $ is not normal.In this paper we prove that $T_{m,n} $ is continuous if and only if $r_{m,n} $ has defect zero. We also prove a convergence result for Pade approximants in the case where the defect is positive.