On the variation-diminishing property
type: Item
instance of: scholarly article
title: On the variation-diminishing property
zbMATH Open document ID: 0907.41020
DOI: 10.1007/BF03322074
author: Ioan Gavrea
author: Heiner Gonska
author: Daniela Kacsó
published in: Results in Mathematics
publication date: 1999-03-02T00:00:00Z
review text: This paper considers a new approach in proving the so-called strong variation-diminishing property for positive linear operators. Further it presents applications for the Durrmeyer operators with Jacobi weights \(t^\alpha (1-t)^\beta\), \(\alpha, \beta>-1\) and also for the family of operators \(P_n\) introduced by the reviewer (\textit{D. H. Mache}, Gewichtete Simultanapproximation in der \(L_p\)-Metrik durch das Verfahren der Kantorovič Operatoren, Dissertation, Univ. Dortmund, 1991]) which describes the link between these Durrmeyer operators with Jacobi weights and the (classical) Bernstein operators. Finally in section 4 the authors prove the relationship between the strong variation-diminishing property and the preservation of convexity, for which ``it follows that the classical Durrmeyer operator, their modifications and the Mache's operator preserve the convexity of orders \(0,\dots,n\).'' In the introduction one can find also a historical remark for the variation-diminishing property.
Mathematics Subject Classification ID: 41A36
zbMATH DE Number: 1140110
zbMATH Keywords: variation-diminishing property
zbMATH Keywords: higher order convexity
zbMATH Keywords: Durrmeyer operators
zbMATH Keywords: Mache operators
reviewed by: Detlef H. Mache
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf03322074
OpenAlex ID: W2051024458
description: scientific article; zbMATH DE number 1140110