Vol. 109, No. 2, 1983

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Bernstein-like polynomial approximation in higher dimensions

Lester Eli Dubins

Vol. 109 (1983), No. 2, 305–311

Let C(K) be the Banach space of continuous, real-valued functions defined on a compact, Hausorff space, K, let 𝒫 = 𝒫(K) be the positive linear forms, P, defined on C(K), for which Pf supf(k) (k K), f C(K), and endow 𝒫(K) with the weak-star topology in which it is, of course, compact. (As is well known, 𝒫 can be identified with the set of countably additive probability measures defined on the Baire subsets of K.) Let P be the power probability on K, the product of denumerable number of copies of K. Then, for each f C(K), the integral of f with respect to P is plainly continuous in P. As Theorem 2 below states, there are no other continuous real-valued functions of P. The proof of this assertion requires a generalization of Bernstein’s version of the celebrated polynomial approximation theorem of Weierstrass, which generalization is provided by Theorem 1.

Mathematical Subject Classification 2000
Primary: 41A65
Received: 19 November 1981
Published: 1 December 1983
Lester Eli Dubins