An extension to rational functions of a theorem of J. L. Walsh on differences of interpolating polynomials

Abstract

Abstract : In this paper, a theorem of J. L. Walsh, on differences of polynomials interpolating in the roots of unity and in the origin, is extended to differences of rational functions interpolating in more general sets. The original result of Walsh is described. Our generalization of Walsh's theorem is in two directions. First, we show that an analogous overconvergence holds for differences of rational interpolants to meromorphic functions F(z). Second, we show that the defining interpolation points can be considerably more general than the roots of unity and the origin. Finally, several concrete examples of our generalization are given, one consisting in applications of Faber polynomials.