?-Bernstein-Kantorovich operators

Abstract

In this article, we introduce a modified class of Bernstein-Kantorovich operators dependingonan integrable function psi(alpha) and investigate their approximation properties. By choosing an appropriate function f, the order of approxi-mation of our operators to a function psi(alpha) is at least as good as the classical Bernstein-Kantorovich operators on the interval[0,1]. We compared the operators defined in this study not only with Bernstein-Kantorovich operators butalso with some other Bernstein-Kantorovich type operators. In this paper, wealso study the results on the uniform convergence and rate of convergenceof these operators in terms of the first- and second-order moduli of continuity, and we prove that our operators have shape-preserving properties. Finally,some numerical examples which support the results obtained in this paper areprovided.