On the rate of convergence of the generalized Durrmeyer type operators for functions of bounded variation (Q1969966)

For a function \(f:[0,1]\to \mathbb{R}\) and \(\alpha\geq 1\), the Durrmeyer-Bézier operator \(D_{n,\alpha}\) is defined by \(D_{n,\alpha} (f;x)= (n+1) \sum^{k=n}_{k=0} Q^\alpha_{n,k} (x)\int^1_0 f(t)p_{n,k}(t) dt\), where \[ p_{n,k}(x) ={n\choose k}x^k(1-x)^{n-k},\;Q^\alpha_{n,k}(x)= \left(\sum^n_{j=k} p_{n,j}(x) \right)^\alpha- \left( \sum^n_{j=k+1}p_{n,j} (x)\right)^\alpha. \] In this paper the authors estimate the rate of convergence of the Durrmeyer-Bézier operators for functions of bounded variation and prove that the sequence \((D_{n,\alpha} (f;x):n \in\mathbb{N})\) converge to the limit \((\alpha+1)^{-1}f(x+)+ \alpha (\alpha+1)^{-1} f(x-)\) for functions of bounded variation \(f\).