Optimal Hermite-Fejér interpolation of algebraic polynomials and the best one-sided approximation on the interval [−1,1

Abstract

In this paper, we give the relations of the optimal Hermite-Fejér interpolation and the best one-sided approximation to the smooth function classes BC2r,+, r∈N in weighted space L1,ω([−1,1]), with a positive, continuous and integrable weight function ω on (−1,1). We proved that the Hermite-Fejér interpolation based on the set of the zeros of some orthogonal polynomials is optimal respectively in the space L1,ω([−1,1]) and the space L∞([−1,1]) and gave the exact constants of the approximation errors of these Hermite-Fejér interpolation. We also obtained the exact constant of the approximation errors of the optimal Hermite-Fejér interpolation when the endpoints of the interval [−1,1] are included in the considered sets of the nodes.