Fundamental Seminorms

Abstract

This chapter discusses fundamental seminorms. The chapter considers a general problem of polynomial approximation in euclidean n-dimensional space. The subject of polynomial approximation was initiated in 1885 with the first version of the Weierstrass theorem for uniform approximation on compact sets of euclidean space. The non-uniform approximation problem on the whole space was initiated with the Bernstein paper and continued to be developed in the so called Bernstein problem. Classically this problem has been studied from the point of view of continuous and integrable functions in the natural context of weighted spaces. The chapter uses the approach in unifying notion of fundamental seminorm in considering a polynomial approximation problem that contains all the cases, mentioned in the chapter of Bernstein's problem. Further, this approach puts in focus the seminorm point of view in approximation theory that has been undertaken for study.