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

TL;DR

This paper proves that convex functions on multi-dimensional space can be approximated by convex polynomials under Freud weights, extending known one-dimensional results to higher dimensions and various norms using a new approach.

Contribution

It extends convex polynomial approximation results with Freud weights from one dimension to multiple dimensions and different Lp norms, employing a novel method.

Findings

Convex functions can be approximated by convex polynomials in weighted norms.

The approximation holds for all dimensions $d \\geq 1$ and for various $p$ norms.

The approach differs significantly from previous methods used in one-dimensional cases.

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 $fW_\alpha\in L_p(R^d)$ 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 $\|(f-P_n)W_\alpha\|_{L_p(R^d)}\to0$ 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.