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Convex polynomial approximation in $R^d$ with Freud weights (1408.2131v3)
Published 9 Aug 2014 in math.CA
Abstract: We show that for multivariate Freud-type weights $W_\alpha(x)=\exp(-|x|\alpha)$, $\alpha>1$, any convex function $f$ on $Rd$ satisfying $fW_\alpha\in L_p(Rd)$ 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 $Rd$ such that $|(f-P_n)W_\alpha|_{L_p(Rd)}\to0$ 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.