Optimal interpolation of convergent algebraic series

Abstract

Let $n\in {\mathbb{N}}$, –1<x ₁<...<x _n <1. Denote $W:=\bigl\{\sum_{r=0}^{\infty} a_r t^r:\ |a_r|\leq 1,\ r\geq n \bigr\}$, t∈(–1,1). Given a function f∈W we try to recover f(ζ) at fixed point ζ∈(–1,1) by an algorithm A on the basis of the information f(x ₁),...,f(x _n ). We find the intrinsic error of recovery $E(W,I):=inf_{A:{\mathbb{R}}^n\to {\mathbb{R}}} \sup_{f\in W} |f(\zeta)-A(If)|$.