Two families of Bernstein–Durrmeyer type operators

This paper deals with two families of Bernstein–Durrmeyer type operators that arise as integral modifications respectively from the classical Bernstein operators and from Bernstein-type operators based on the Polya distribution. An important role is played by the so-called genuine elements of each class, the only ones that reproduce linear functions. Asymptotic formulae and direct convergence results are stated. Finally some modifications of the sequences of the families are introduced, in such a way that the resulting operators reproduce linear functions and are comparable with the genuine sequences.