Convex polynomial approximation in $R^d$ with Freud weights

Abstract

We show that for multivariate Freud-type weights $W_{\alpha }(x)=\exp (-|x{|}^{\alpha })$, $\alpha >1$, any convex function $f$ on $R^d$ satisfying $f{W}_{\alpha }\in L_p(R^d)$ if $1\le p<\infty$, or ${\lim }_{|x|\to \infty }f(x)W_{\alpha }(x)=0$ if $p=\infty$, can be approximated in the weighted norm by a sequence $P_n$ of algebraic polynomials convex on $R^d$ such that $\Vert (f-P_n)W_{\alpha }{\Vert }_{L_p(R^d)}\to 0$ as $n\to \infty$. This extends the previously known result for $d=1$ and $p=\infty$ obtained by the first author to higher dimensions and integral norms using a completely different approach.