Asymptotic Expansion of the Multivariate Bernstein Polynomials on a Simplex

Abstract

In this note we study the local behaviour of the multi-variate Bernstein polynomials on a d-dimensional simplex S. For functions f admitting derivatives of sufficient high order in a point x of S we derive the complete asymptotic expansion of the Bernstein polynomials of order n as n tends to infinity. All the coefficients of in the asymptotic expansion, which only depend on f and x, are calculated explicitly. It turns out that combinatorial numbers play an important role. Our results generalize recent formulae due to R. Zhang.