Rate of convergence of Baskakov-Bézier type operators for locally bounded functions.
Let \(b_{n,k}(x)= {n+k-1\choose k} x^k (1+x)^{-n-k}\) and \(J_{n,k}(x)=\sum\limits_{j=k}^{\infty} b_{n,j}(x), \alpha\geq 1\). The Baskakov-Bézier operator \(B_{n,\alpha}\) is defined by \[ B_{n,\alpha}(f,x)=\sum\limits_{k=0}^{\infty}f(\frac{k}{n})[J_{n,k}^\alpha(x)-J_{n,k+1}^\alpha(x)]. \] In this paper the authors studied the rate of convergence of the operator \(B_{n,\alpha}\) for locally bounded functions by using some results of probability theory. The estimate is essentially the best possible. The method is not new (cf. \textit{S. Guo} and \textit{M. K. Khan} [J. Approximation Theory 58, No.1, 90--101 (1989; Zbl 0683.41030)]).
scientific article; zbMATH DE number 2017159