A New Estimate on the Rate of Convergence of Durrmeyer-Bézier Operators

Recommended by Vijay Gupta We obtain an estimate on the rate of convergence of Durrmeyer-Bézier operaters for functions of bounded variation by means of some probabilistic methods and inequality techniques. Our estimate improves the result of Zeng and Chen 2000. In 2000, Zeng and Chen 1 introduced the Durrmeyer-Bézier operators D n,α which are defined as follows: D n,α f, x n 1 n k0 Q α nk x 1 0 ftp nk tdt, 1.1 where f is defined on 0, 1, α ≥ 1, Q α nk x J α nk x − J α n,k1 x, J nk x n jk p nj x, n k0 p nk x 1 0 ftp nk tdt. 1.2 Concerning the approximation properties of operators D n,1 f and some results on approximation of functions of bounded variation by positive linear operators, one can refer