Positive convergent approximation operators associated with orthogonal polynomials for weights on the whole real line
type: Item
instance of: scholarly article
title: Positive convergent approximation operators associated with orthogonal polynomials for weights on the whole real line
zbMATH Open document ID: 0604.41025
DOI: 10.1016/0021-9045(86)90060-2
author: Arnold Knopfmacher
published in: Journal of Approximation Theory
publication date: 1986-01-01T00:00:00Z
review text: 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].
Mathematics Subject Classification ID: 41A36
Mathematics Subject Classification ID: 42C05
zbMATH DE Number: 3975534
zbMATH Keywords: interpolation operators
zbMATH Keywords: weights
zbMATH Keywords: Gaussian quadrature
reviewed by: H. R. Dowson
MaRDI profile type: Publication
full work available at URL: 0021-9045(86)90060-2
OpenAlex ID: W2057229610
cites work: A class of orthogonal polynomials
cites work: Q4771257
cites work: On Markov-Bernstein-type inequalities and their applications
cites work: Gaussian quadrature, weights on the whole real line and even entire functions with nonnegative even order derivatives
cites work: Orthogonal polynomials
cites work: Exact bounds for orthogonal polynomials associated with exponential weights
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description: scientific article; zbMATH DE number 3975534