Modified Szász-Kantorovich operators with better approximation

In this paper, we introduce a new family of Sz?sz-Mirakjan Kantorovich type operators Kn,?(f;x), depend on a function ? which satisfies some conditions. In this way we obtain all moments and central moments of the new operators in terms of two numbers M1,? and M2,?, which are integrals of ? and ?2, respectively. This is a new approach to have better error estimation, because in the case of Kn,?(1;x) = 1, the order of approximation to a function f by an operator Kn,?(f;x) is more controlled by the term Kn,?((t?x)2;x). Since the different functions ? gives different values for M1,? and M2,?, it is possible to search for a function ? with different values of M1,? and M2,? to make Kn,?((t?x)2; x) smaller. By using above approach, we show that there exist a function ? such that the operator Kn,?(f;x) has better approximation then the classical Sz?sz-Mirakjan Kantorovich operators. We obtain some direct and local approximation properties of new operators Kn,?(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 x2 and we show that King-Type generalizations of Kn,?(f;x) has better approximation properties than Kn,?(f;x) and than the classical Sz?sz-Mirakjan-Kantorovich operator. Furthermore, we illustrate approximation results of these operators graphically and numerically.