The Genuine Bernstein–Durrmeyer Operators and Quasi-Interpolants

Abstract

In this paper we consider the so-called genuine Bernstein–Durrmeyer operators and define corresponding quasi-interpolants of order \({r \in \mathbb{N}_0}\) in terms of certain differential operators. These quasi-interpolants preserve all polynomials of degree at most r + 1. We analyse the eigenstructure of the differential operators and the quasi-interpolants and prove as main results an error estimate of Jackson–Favard type for sufficiently smooth functions and an upper bound for the error of approximation in the sup-norm in terms of an appropriate K-functional.