Asymptotic expansions for multivariate polynomial approximation

In this paper the approximation of multivariate functions by (multivariate) Bernstein polynomials is considered. Building on recent work of Lai (J. Approx. Theory 70 (1992) 229–242), we can prove that the sequence of these Bernstein polynomials possesses an asymptotic expansion with respect to the index n. This generalizes a corresponding result due to Costabile et al. (BIT 36 (1996) 676–687) on univariate Bernstein polynomials, providing at the same time a new proof for it. After having shown the existence of an asymptotic expansion we can apply an extrapolation algorithm which accelerates the convergence of the Bernstein polynomials considerably; this leads to a new and very efficient method for polynomial approximation of multivariate functions. Numerical examples illustrate our approach.