On uniform convergence of rational, Newton-Pad� interpolants of type (n, n) with free poles asn??

Abstract

Letf be meromorphic in ?. We show that there exists a sequence of distinct interpolation points {z j } j=1 ? , and forn?1, rational functions,R n (z) of type (n, n) solving the Newton-Pade (Hermite) interpolation problem, $$R_n (z_j ) = f(z_j ), j = 1,2,...2n + 1,$$ and such that for each compact subsetK of ? omitting poles off, we have $$\mathop {\lim }\limits_{n \to \infty } ||f - R_n ||_{L\infty (K)}^{1/n} = 0.$$ Extensions are presented to the case wheref(z) is meromorphic in a given open set with certain additional properties, and related results are discussed.