Convex polynomial approximation in [formula omitted] with Freud weights

Abstract

We show that for multivariate Freud-type weights Wα(x)=exp(−|x|α), α>1, any convex function f on Rd satisfying fWα∈Lp(Rd) if 1≤p<∞, or lim|x|→∞f(x)Wα(x)=0 if p=∞, can be approximated in the weighted norm by a sequence Pn of algebraic polynomials convex on Rd such that ‖(f−Pn)Wα‖Lp(Rd)→0 as n→∞. This extends the previously known result for d=1 and p=∞ obtained by the first author to higher dimensions and integral norms using a completely different approach.