The Choquet boundary and approximation theory

Abstract

Many of the classical theorems in approximation theory can be formulated in terms of convergence of a sequence of linear operators to the identity operator. To illustrate this, we consider one example in detail, obtaining Bernstein's proof of the Weierstrass approximation theorem. For each n ≥ 1, define the operator B _n from C[0,1] into the polynomials of degree at most n by setting, for f ∈ C[0,1].