Extrapolation Properties of Multivariate Bernstein Polynomials

Abstract

We consider some connections between the classical sequence of Bernstein polynomials and the Taylor expansion at the point 0 of a $C^\infty$ function f defined on a convex open subset $\Omega \subset \mathbb {R}^d$ containing the d-dimensional simplex $S^d$ of $\mathbb {R}^d$. Under general assumptions, we obtain that the sequence of Bernstein polynomials converges to the Taylor expansion and hence to the function f together with derivatives of every order not only on $S^d$ but also on the whole $\Omega$. This result yields extrapolation properties of the classical Bernstein operators and their derivatives. An extension of the Voronovskaja’s formula is also stated.