Generalized Bernstein-Durrmeyer operators and the associated limit semigroup
In recent years there have been quite a few generalizations of the Bernstein-Durrmeyer operators either by shifting the exponents in the Bernstein polynomials in various ways, or by shifting the abscissae where the function is evaluated in various ways. The current paper is an attempt to put many of the generalizations under one roof. The author applies the Lototsky--Schnabl operators to a generalization of the Beta operators, originally introduced by Lupaş. Investigating separately properties of the two operators the author proves some of the classical error estimates involving the first and (sometimes) the second moduli of smoothness of the function. Also he is able to obtain a Voronovskaja-type result for the generalized operators. The last part of the paper is devoted to the investigation of the relations between the new operators and the associated semigroup and its infinitesimal generator.
scientific article; zbMATH DE number 1346404
scientific article