Polynomial approximation in several variables

Abstract

This paper is about the characterization of the rate of best polynomial approximation on a domain in Rd by multivariate polynomials of degree n via a suitable modulus of continuity or smoothness. We shall be concerned with the situations: approximation on convex polytopes and approximation on (not necessarily convex) smooth algebraic domains. We shall introduce appropriate moduli of smoothness and prove the corresponding Jackson theorem. The usual weak converses are also true.