Bernstein-like polynomial approximation in higher dimensions
Let K be a compact Hausdorff space and \({\mathcal C}(K)\) the Banach space of real continuous functions on K. Let \({\mathcal P}\) be the set of positive linear forms P defined on \({\mathcal C}(K)\) for which \(Pf\leq\sup \{f(k):k\in K\},\) and let \({\mathcal P}\) be endowed with the weak-star topology. Let \(p^{\infty}\) be the power probability on \(K^{\infty}\). In the main theorem of the paper it is shown that for each \(f\in {\mathcal C}(K^{\infty})\) the integral of f with respect to \(P^{\infty}\) is the only continuous function of \({\mathcal P}\). In order to prove this theorem a generalization of Bernstein's version of the polynomial approximation theorem of Weierstrass is also given.
scientific article; zbMATH DE number 3869919