Asymptotic expansions for multivariate polynomial approximation

Abstract

In this paper the approximation of multivariate functions by (multivariate) Bernstein polynomials is considered. Building on recent work of Lai [3], 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, Gualtieri and Serra [2] 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.

Asymptotic expansion of multivariate Kantorovich type operators

“…They satisfy the conditions of the theorem A for ϕ = t and φ(n) = n. As a consequence of that, in [9] (see also the results of Lai and Walz [6,11] for the non-differentiated sequence) it is proved that given r ∈ N even, k ∈ N m 0 , x ∈ and f ∈ C( ) differentiable at x up to the order r + |k|,…”

“…The existence of an asymptotic formula of a sequence allows to study the order of convergence, saturation results and make possible to use extrapolation algorithms to improve the convergence (see, for instance, [4,10,11]). …”

Iterated Bernstein operators for distribution function and density estimation: Balancing between the number of iterations and the polynomial degree

“…Consider now the restriction B n of • B n to P n . Since B n reproduces only the linear polynomials (even quadratic polynomials are not reproduced by B n 8 -see Walz (2000)), the operator G n := 1 − B n is injective. All its eigenvalues are positive, and the maximal one is 1 Impens (2003)).…”

Recent Progress on Univariate and Multivariate Polynomial and Spline Quasi-interpolants

“…Numerical applications are still not very much developed. However, QIs can be useful in approximation and estimation [21][22] [85], in numerical quadrature [23][73][75], and for the numerical solution of integral or partial differential equations.…”