On the rate of convergence of the Durrmeyer operator for functions of bounded variation (Q1116389)

Let f be a function defined on [0,1]. The Durrmeyer operator \(M_ n\) applied to f is determined in terms of \[ M_ n(f,x)=(n+1)\sum^{n}_{k=0}p_{nk}(x)\int^{1}_{0}f(t)p_{nk}(t)dt \] where \(p_{nk}(x)=\left( \begin{matrix} n\\ k\end{matrix} \right)x^ k(1- x)^{n-k}\). An estimate for the rate of convergence of the Durrmeyer operator is given for functions of bounded variation at each point of (0,1). To this end properties of the kernel \(p_{nk}(x)\) and some results of probability are used. The function \(f(t)=| t-x|\) \((0<x<1)\) taken as an example shows that the estimate is essentially the best possible.