Positive convergent approximation operators associated with orthogonal polynomials for weights on the whole real line
Positive interpolation operators \({\mathcal J}_{n,p}\), where \(0<p<\infty\), defined by \[ {\mathcal J}_{n,p}[f](x)=\frac{\sum {n}{k=1}\lambda_{kn}f(x_{kn})| K_ n(x,x_{kn})|^ p}{\sum^{n}_{k=1}\lambda_{kn}| K_ n(x,x_{kn})|^ p} \] for weights \(W^ 2(x)=\exp (-2Q(x))\), are introduced. Here \(K_ n(x,t)\) is the kernel of degree at most n-1 in x, t for the partial sums of the orthogonal expansions with respect to \(W^ 2\), and \(\{x_{kn}\}\) and \(\{\lambda_{kn}\}\) are the abcissas and weights in the Gaussian quadrature of order n. Their basic properties are established, and their convergence is proved for \(1<p\leq 2\) and a certain class of weights on the whole real line. \textit{P. G. Nevai} [Orthogonal polynomials, Mem. Am. Math. Soc. 213 (1979; Zbl 0405.33009)] has considered the special case \(p=2\) and weights on [-1,1].
scientific article; zbMATH DE number 3975534
scientific article