The Eigenstructure of Operators Linking the Bernstein and the Genuine Bernstein–Durrmeyer operators

Abstract

We study the eigenstructure of a one-parameter class of operators $${U_{n}^{\varrho}}$$ of Bernstein–Durrmeyer type that preserve linear functions and constitute a link between the so-called genuine Bernstein–Durrmeyer operators U n and the classical Bernstein operators B n . In particular, for $${\varrho\rightarrow\infty}$$ (respectively, $${\varrho=1}$$ ) we recapture results well-known in the literature, concerning the eigenstructure of B n (respectively, U n ). The last section is devoted to applications involving the iterates of $${U_{n}^{\varrho}}$$ .