Weighted Hermite-Fejér interpolation on Laguerre nodes
Let \(w\) be a Laguerre weight function on the halfline \((0,\infty)\). The author introduces weighted Hermite-Fejér interpolation polynomials \(H_{w,n}\) satisfying \(wH_{w,n}=wf\) and \((wH_{w,n})'=0\) at the nodes \(x_1,\dots,x_n\) of a point system. The following convergence result is proved: Let \(\alpha >0\) and \(w(x)=x^\alpha e^{-x}\). If the continuous function \(f\) satisfies \(w(x)f(x) \to 0\) as \(x\to 0\) or \(x\to \infty\), and \(x_1,\dots,x_n\) are the roots of the Laguerre polynomials with parameter \(\alpha -\rho\) for some \(\rho\) with \(0 < \rho <1\) then \(wH_{w,n}\) converges to \(wf\) as \(n \to \infty\) uniformly on the halfline \((0,\infty)\).
scientific article; zbMATH DE number 2099110
scientific article