Local approximation by a variant of Bernstein–Durrmeyer operators

This paper deals with the local approximation properties of a certain variant M˜n of the Bernstein–Durrmeyer operators. Firstly, we obtain an estimate on the rate of convergence of M˜n by means of the decomposition technique for functions of bounded variation. It will be shown that our estimate in its most convenient form (Corollary 2) is asymptotically optimal. Furthermore, we derive the complete asymptotic expansion for the sequence of the operators M˜n as nn tends to infinity.