Convex polynomial approximation in \(\mathbb{R}^d\) with Freud weights (Q2256607)

On \(\mathbb{R}^d\) consider the Freud weight \(W_{\alpha}({\mathbf x})=e^{-|{\mathbf x}|^{\alpha}}\), \(\alpha>1\), where \({\mathbf x}=(x_1,\ldots,x_d)\in\mathbb{R}^d\), \(|{\mathbf x}|=(x_1^2+\ldots+x_d^2)^{1/2}\). Denote \(C_{W_{\alpha}}(\mathbb{R}^d):=\{f\in C(\mathbb{R}^d),\;\lim_{|{\mathbf x}|\to\infty}W_{\alpha}({\mathbf x})f({\mathbf x})=0\}\). For \(1\leq p\leq\infty\), denote by \(L_p(\mathbb{R}^d)\) the space of measurable functions \(f:\mathbb{R}^d\to\mathbb{R}\), such that \(\| f\|_{L_p(\mathbb{R}^d)}<\infty\), where \(\| f\|_{L_p(\mathbb{R}^d)}=(\int_{\mathbb{R}^d}|f({\mathbf x})|^pd{\mathbf x})^{1/p}\), for \(1\leq p<\infty\) and \(\| f\|_{L_{\infty}(\mathbb{R}^d)}=\) ess\ sup\(_{{\mathbf x}\in\mathbb{R}^d}|f({\mathbf x})|\), for \(p=\infty\). Let \(\Pi_{n,d}^{(2)}\) be the space of convex polynomials on \(\mathbb{R}^d\), with the total degree \(\leq n\). The main result is the following. If \(f:\mathbb{R}^d\to \mathbb{R}\) is convex on \(\mathbb{R}^d\) and \(fW_{\alpha}\in L_p(\mathbb{R}^d)\), for \(1\leq p<\infty\) and \(f\in C_{W_{\alpha}}(\mathbb{R}^d)\), for \(p=\infty\), then \[ \inf_{P\in \Pi_{n,d}^{(2)}}\| (f-P)W_{\alpha}\|_{L_p(\mathbb{R}^d)}=0,\;n\to\infty. \]