On approximation properties of some non-positive Bernstein-Durrmeyer type operators modified in the Bezier-King sense

In this paper we propose some new Bernstein-Durrmeyer type operators modified in Bezier-King sense, which are not positive on the entire interval [0, 1] . We prove that, even though the operators are not positive on the entire [0, 1], they can approximate all continuous functions on [0, 1], first, by using the first order modulus of continuity, and then the second order one, with the appropriate K-functionals. Finally, we prove a Voronovkaja type theorem.