On the rates of convergence of Bernstein-Chlodovsky polynomials and their Bezier-type variants

In this article, we consider the Chlodovsky polynomials Cn f and their Bezier variants Cn, f, with 0, for locally bounded functions f on the interval [0, ). Using the Chanturiya modulus of variation we give estimates for the rates of convergence of Cn f (x) and Cn, f (x) at those points x 0 at which the one-sided limits f (x+), f (x-) exist. The recent results of Karsli and Ibiki [H. Karsli and E. Ibikli, Rate of convergence of Chlodovsky type Bernstein operators for functions of bounded variation, Numer. Funct. Anal. Optim. 28(3-4) (2007), pp. 367-378; H. Karsli and E. Ibikli, Convergence rate of a new Bezier variant of Chlodovsky operators to bounded variation functions, J. Comput. Appl. Math. 212(2) (2008), pp. 431-443.] are essentially improved and extended to more general classes of functions.