Summability of sequences of interpolatory polynomials in the roots of unity: rate of convergence

Abstract

Let be a region containing the disk for some R>1, and let f be a function holomorphic in G. Furthermore, let L n(.;f) denote the Lagrange interpolatory polynomial of f in the (n + 1)st roots of unity. Then it is well-known that L n(z;f)→ f(z) (n→∞) locally uniformly in D R. In [2] and [3] we applied certain matrix summability methods to the sequence L n(.;f) in order to enlarge the set of convergence. This set is an open set, it depends on the summability method and the singularities of f, and it often contains the disk D R as a proper subset. The aim of this paper is to study the rate of convergence.