Modified Szasz-Kantorovich operators with better approximation

Abstract

In this paper, we introduce a new family of Szasz-Mirakjan Kantorovich type operators K-n,K-psi(f; x), depend on a function psi which satisfies some conditions. In this way we obtain all moments and central moments of the new operators in terms of two numbers M-1,M-psi and M-2,M-psi, which are integrals of psi and psi(2), respectively. This is a new approach to have better error estimation, because in the case of K-n,K-psi(1; x) = 1, the order of approximation to a function f by an operator K-n,K-psi( f ; x) is more controlled by the term Kn,psi((t-x)2; x). Since the different functions psi gives different values for M1,psi and M-2,M- psi, it is possible to search for a function psi with different values of M-1,M-psi and M-2,M- psi to make K-n,K-psi((t - x)(2); x) smaller. By using above approach, we show that there exist a function psi such that the operator K-n,K-psi(f; x) has better approximation then the classical Szasz-Mirakjan Kantorovich operators. We obtain some direct and local approximation properties of new operators K-n,K-psi(f; x) and we prove that our new operators have shape preserving properties. Moreover, we also introduced two different King-Type generalizations of our operators, one preserving x and the other preserving x(2) and we show that King-Type generalizations of K-n,K-psi(f; x) has better approximation properties than K-n,K-psi( f ; x) and than the classical Szasz-Mirakjan-Kantorovich operator. Furthermore, we illustrate approximation results of these operators graphically and numerically.