Uniform and mean approximation by certain weighted polynomials, with applications (Q1103796)

Let \({\mathcal P}_ n\) be the class of polynomials of degree \(\leq n\). Let \(VSF(\alpha)\) be the class of functions \(w(x)=\exp (-Q(x))\) where Q(x) is even, continuous and \(Q'(x)\) exists for \(x>0\), while xQ'(x) remains bounded as \(x\to 0_+\). Further, assume that \(Q'''(x)\) exists for x large enough, and for some \(C>0\) and \(\alpha >0,Q'(x)>0\), x large enough, \(x^ 2| Q'''(x)| /Q'(x)\leq C,\) x large enough, and \(\lim_{x\to \infty}(1+xQ''(x)/Q'(x))=\alpha.\) The main results are: Let \(W\in VSF(\alpha)\), \(\alpha >0\) and \(a_ n=a_ n(W)\) be positive roots of the equation \(n=(2/\pi)\int^{1}_{0}a_ nxQ'(x)/\sqrt{1-x^ 2}\cdot dx,\) such that \(\limsup_{n\to \infty}a_ n/n>0.\) Let g(x) be continuous in \({\mathbb{R}}\) and \(g(0)=0\). Let \(\{k_ n\}_ 1^{\infty}\) be a sequence of nonnegative integers with \(\lim_{n\to \infty}k_ n/n=0\). Then, there exists \(P_ n\in {\mathcal P}_{n-k_ n}\), \(n=1,2,..\). such that \(\lim_{n\to \infty}\| g(x)-P_ n(x)W(a_ nx)\|_{L_{\infty}({\mathbb{R}})}=0\) if and only if \(g(x)=0\) for \(| x| \geq 1\). A similar result holds for \(L_ p({\mathbb{R}})\). These results confirm a conjecture of Saff for \(W(x)=\exp (-| x|^{\alpha})\) when \(\alpha >1\). Other applications of these results are upper bounds for Christoffel functions.