Extrapolation Properties of Multivariate Bernstein Polynomials

We consider some connections between the classical sequence of Bernstein polynomials and the Taylor expansion at the point 0 of a C-infinity function f defined on a convex open subset Omega subset of R-d containing the d-dimensional simplex S-d of 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.