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

Oleksandr Maizlish; Andriy Prymak

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arXiv · 1408.2131

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

Oleksandr Maizlish·Andriy Prymak

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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.

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BibTeX RIS

Oleksandr Maizlish, Andriy Prymak. 2014-08-09. Convex polynomial approximation in $R^d$ with Freud weights. https://arxiv.org/abs/1408.2131

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