Convex polynomial approximation in RdRd with Freud weights

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