A generalized Bernstein approximation theorem

Abstract

A celebrated theorem of Weierstrass states that any continuous real-valued function f defined on the closed interval [0, 1] ⊂ ℝ is the limit of a uniformly convergent sequence of polynomials. One of the most elegant and elementary proofs of this classic result is that which uses the Bernstein polynomials of f one for each integer n ≥ 1. Bernstein's Theorem states that B n ( f ) → f uniformly on [0, 1] and, since each B n ( f ) is a polynomial of degree at most n , we have as a consequence Weierstrass' theorem. See for example Lorentz [ 9 ]. The operator B n , defined on the space C ([0, 1]; ℝ) with values in the vector subspace of all polynomials of degree at most n has the property that B n ( f ) ≥ 0 whenever f ≥ 0. Thus Bernstein's Theorem also establishes the fact that each positive continuous real-valued function on [0, 1] is the limit of a uniformly convergent sequence of positive polynomials. This raises the following natural question: consider a compact Hausdorff space X and the convex cone C + ( X ):= { f ∈ C ( X ; ℝ); f ≥ 0}. Now the analogue of Bernstein's Theorem would be a theorem stating when a convex cone contained in C + ( X ) is dense in it. More generally, one raises the question of describing the closure of a convex cone contained in C ( X ; ℝ), and, in particular, the closure of A + := { f ∈ A ; f ≥ 0}, where A is a subalgebra of C ( X ; ℝ).