Local approximation by a variant of Bernstein–Durrmeyer operators Original Research Article

This paper deals with the local approximation properties of a certain variant View the MathML sourceM˜n of the Bernstein–Durrmeyer operators. Firstly, we obtain an estimate on the rate of convergence of View the MathML sourceM˜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 View the MathML sourceM˜n as nn tends to infinity.