On Multivariate Incomplete Polynomials on Starlike Domains

Abstract

Multivariate incomplete polynomials are considered on compact 0-symmetric starlike domains. Problems of density and quantitative approximation properties of such polynomials are investigated. It is shown that density holds for a certain class of starlike domains which includes both convex and some nonconvex domains. On the other hand, a family of nonconvex starlike domains is also found for which density fails. In addition, it is also shown that on 0-symmetric convex bodies in \(\mathbb{R}^{d}\), continuous functions can be approximated by θ-incomplete polynomials with the rate O(ω2(n−1/(d+3))). Moreover, if the convex body is the intersection of simplexes with vertex at the origin, then this order improves to \(O (\omega_{2}(f,1/\sqrt{n}) )\).