Mediterr. J. Math. (2019) 16:109 https://doi.org/10.1007/s00009-019-1392-0 c  Springer Nature Switzerland AG 2019 Extrapolation Properties of Multivariate Bernstein Polynomials Michele Campiti and Ioan Ra¸sa Abstract. We consider some connections between the classical sequence of Bernstein polynomials and the Taylor expansion at the point 0 of a C ∞ function f defined on a convex open subset Ω ⊂ R d containing the d-dimensional simplex S d of R d . Under general assumptions, we ob- tain that the sequence of Bernstein polynomials converges to the Taylor expansion and hence to the function f together with derivatives of ev- ery order not only on S d but also on the whole Ω. This result yields extrapolation properties of the classical Bernstein operators and their derivatives. An extension of the Voronovskaja’s formula is also stated. AMS Classification. 41A10, 41A28, 41A36, 41A63. Keywords. Bernstein operators, Extrapolation, Voronovskaja’s formula, Taylor series. 1. Extrapolation Properties on a Compact Subset of R d The problem of the convergence of Bernstein polynomials outside the interval [0, 1] for suitable functions has been posed from many years and a conver- gence property is substantially well known (see e.g., [6, pp. 204–206] or more recently [3, p. 268, Exercise 4.7]). The aim of this paper is, on one hand, to deepen this property, and on the other hand, to extend it in more general contexts. This is also motivated by some recent researches where values of Bernstein operators outside the interval [0, 1] are explicitly required (see, e.g., [5, (3.4) with m = n and p = B n (f )]); moreover, the extension of the Voronovskaja formula outside the interval [0, 1] (and outside the simplex S d in the multi-dimensional setting) opens the possibility of studying new connections with suitable differential operators and evolution problems (in this context, we refer to [2, Chapter 6] for an exhaustive treatment). For the sake of brevity, we consider directly the multivariate case of Bernstein operators on a d-dimensional simplex. M. Campiti: Work performed under the auspices of G.N.A.M.P.A. (I.N.d.A.M.). 0123456789().: V,-vol