Bernstein-like polynomial approximation in higher dimensions
type: Item
instance of: scholarly article
title: Bernstein-like polynomial approximation in higher dimensions
zbMATH Open document ID: 0546.41023
author: Lester E. Dubins
published in: Pacific Journal of Mathematics
publication date: 1983-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 41A36
Mathematics Subject Classification ID: 28A25
Mathematics Subject Classification ID: 60B05
zbMATH DE Number: 3869919
zbMATH Keywords: power probability
zbMATH Keywords: Weierstrass approximation theorem
zbMATH Keywords: Bernstein-like polynomial approximation
reviewed by: Ganesh Datta Dikshit
MaRDI profile type: Publication
full work available at URL: pjm.1983.109.305
OpenAlex ID: W2055589406
DOI: 10.2140/PJM.1983.109.305
description: scientific article; zbMATH DE number 3869919