Shape-preserving multivariate polynomial approximation in C[−1,1]m

Abstract

We construct multivariate polynomials attached to a function f of m variables, m≥2 , which approximate f with Jackson-type rate involving a multivariate Ditzian-Totik ω2φ-modulus and preserve some natural kinds of multivariate monotonicity and convexity of function. The result extends the bivariate case in Gal (2002), but does not follow straight from it and requires a careful calculation.