Uniform approximation by Nevai operators
type: Item
instance of: scholarly article
title: Uniform approximation by Nevai operators
zbMATH Open document ID: 0999.41009
author: Biancamaria Della Vecchia
published in: Journal of Approximation Theory
publication date: 2002-09-12T00:00:00Z
review text: The author investigate the Nevai operator \(N_n\) interpolating a function \(f\) on \([-1,1]\) in the more general weighted approximation case, when \(f\) may be unbounded at \(\pm 1\). It is shown that, similarly as for polynomials, the weighted convergence for the operators \(N_n\) with Pollaczek type weights is not guaranteed in general. Therefore the author considers weights vanishing algebraically at \(\pm 1\), i.e. functions having an algebraic singularity at \(\pm 1\). For such functions weighted uniform approximation estimates by \(N_n\) involving a suitable modulus of smoothness are given. Converse results are also established. Useful tools for these results are new weighted Markov-Bernstein inequalities for \(N_n\). It is shown that these results are sharp in some sense. When \(f\) is continuous on the whole interval \([-1,1]\), more precise direct and converse results, solving the saturation problem of \(N_n\) for \(s>2\), are obtained. Finally, the difficult problem of saturation of \(N_n\) for \(s\geqslant 2\) is investigated.
Mathematics Subject Classification ID: 41A20
Mathematics Subject Classification ID: 41A36
Mathematics Subject Classification ID: 41A40
Mathematics Subject Classification ID: 41-06
zbMATH DE Number: 1800256
zbMATH Keywords: approximation by rational functions
zbMATH Keywords: approximation by positive operator
zbMATH Keywords: Nevai operator
zbMATH Keywords: generalized smooth Jacobi weight
zbMATH Keywords: Christoffel function
zbMATH Keywords: weighted and unweighted uniform approximation
zbMATH Keywords: weighted uniform convergence
zbMATH Keywords: Pollaczek type weights
zbMATH Keywords: Shepard operator
zbMATH Keywords: weighted norm
zbMATH Keywords: weighted Markov-Bernstein type inequalities
zbMATH Keywords: converse theorems
zbMATH Keywords: saturation
reviewed by: Eleonora A. Storozhenko
MaRDI profile type: Publication
OpenAlex ID: W1997424557
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DOI: 10.1006/JATH.2001.3666
description: scientific article; zbMATH DE number 1800256