Uniform Approximation of Functions Through Optimal Partitioning

Abstract

In this paper, the problem of best uniform polynomial approximation to a continuous function on a compact set X in a Euclidean space is approached through the partitioning of X and the definition of a corresponding norm. If the width of this partition is sufficiently small, the unique best polynomial approximation in the corresponding norm is arbitrarily close to the set of best uniform polynomial approximations. Furthermore, for fixed k greater than the number of points in a critical point set, there exists a partition of X into k subsets, so that the corresponding best polynomial approximation is arbitrarily close to the set of best uniform polynomial approximations.