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. (C) 2000 Elsevier Science B.V. All rights reserved.