On the variation-diminishing property
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.
scientific article; zbMATH DE number 1140110