Best local weighted approximation: an approach with abstract seminorms

Abstract

In this paper, we extend results given by Z´o and Cuenya in 2007 about a general approach to problems of best vector-valued approximation on small regions from a finite dimensional subspace of polynomials of some degree. This approach is called best local approximation. We consider a weighted local approximation of a vector valued function on the origin and a weighted best local approximation of a real valued function on several points, similar to classical problems in best local approximation with balanced neighborhood.