On the Cheney and Sharma operator
type: Item
instance of: scholarly article
title: On the Cheney and Sharma operator
zbMATH Open document ID: 0857.41020
author: José A. Adell
author: Ana Pérez-Palomares
author: Jesús de la Cal
published in: Journal of Mathematical Analysis and Applications
publication date: 1997-02-26T00:00:00Z
review text: The authors consider the double indexed Bernstein power series operator \(P_{a,t}\) introduced by \textit{E. W. Cheney} and \textit{A. Sharma} [Can. J. Math. 16, 241-252 (1964; Zbl 0128.29001)] and related to the classical Laguerre polynomials. Sharp estimates of the first two moments are obtained. These enable the authors to prove that, when acting on a continuous function \(f\), the convergence of \(P_{a,t}f\) to \(f\) is uniform on the whole interval \([0,1]\). Further, it is shown that the rates of convergence depend on the way in which \(t/a\) tends to \(0\). It is also shown that \(P_{a,t}\) preserves monotonicity and global smoothness. Finally, the authors discuss the monotonicity properties of \(P_{a,t}\) with respect to both parameters. A probabilistic approach is adopted throughout.
Mathematics Subject Classification ID: 41A36
zbMATH DE Number: 908597
zbMATH Keywords: Bernstein power series operator
reviewed by: H. R. Dowson
MaRDI profile type: Publication
full work available at URL: jmaa.1996.0230
OpenAlex ID: W2087907981
DOI: 10.1006/JMAA.1996.0230
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description: scientific article; zbMATH DE number 908597