Polynomial approximation and interpolation on the real line with respect to general classes of weights
In this interesting paper, the authors extend results on weighted polynomial approximation and Lagrange interpolation for Freud weights such as \(\exp(-| x| ^\alpha)\), \(\alpha>1\) by allowing the weight to vanish at finitely many points on the real axis. Recall Bernstein's approximation problem: let \(w:\mathbb{R}\to (0,\infty)\) be continuous and let \(C_w\) denote the space of continuous functions \(f:\mathbb{R}\to\mathbb{R}\) with \[ \lim_{| x|\to\infty} (fw)(x)=0 \] and with norm \[ \| f\|:= \sup_{x\in\mathbb{R}} | fw|(x). \] When are the polynomials dense in \(C_w\)? A necessary condition is \[ \int_{-\infty}^\infty \frac{|\log w(x)|} {1+x^2} dx=\infty. \] When \(w\) is even and \(|\log w(e^x)| \) is convex in \([0,\infty)\), this is necessary and sufficient. In particular for \(w(x)= \exp(-| x|^\alpha)\), the polynomials are dense if \(\alpha\geq 1\). The authors discuss the case where \(w\) vanishes (even as strongly as an essential singularity) at finitely many points on the real line. Then, as their main result, they prove that there is an array of Lagrange interpolation points giving a Lebesgue constant of order \(\log n\). This generalizes earlier work of the second author for Freud weights, where \(\log n\) is known to be optimal.
scientific article; zbMATH DE number 1209935