Modified Szasz-Kantorovich Operators With Better Approximation

Abstract

In this paper, we introduced a new family of Sz´asz-Mirakjan Kantorovich
type operators Kn,ψ(f; x), depend on a function ψ which satisfies some
conditions. In this way we obtained all moments and central moments
of the new operators in terms of two numbers M1,ψ and M2,ψ, which
are integrals of ψ and ψ2 respectively. This is a new approach to have
better error estimation because in the case of Kn,ψ(1; x) = 1, the
order of approximation to a function f by an operator Kn,ψ(f; x) is
more controlled by the term Kn,ψ((t − x)2; x). Since the different
functions ψ gives different values for M1,ψ and M2,ψ, it is possible to
search for a function ψ with different values of M1,ψ and M2,ψ to make
Kn,ψ((t − x)2; x) smaller. By using above approach, we show that
there exist a function ψ such that the operator Kn,ψ(f; x) has better
approximation then classical Sz´asz-Mirakjan Kantorovich operators. We
obtain some direct and local approximation properties of new operators
Kn,ψ(f; x) and we prove that our new operators has shape preserving
properties. Moreover, we also introduced two different King-Type
generalizations of our operators, one preserving x and other preserving
x2 and we show that King-Type generalizations of Kn,ψ(f; x) has
better approximation properties from Kn,ψ(f; x) and from the classical
Sz´asz-Mirakjan-Kantorovich operator. Furthermore, we illustrate
approximation results of these operators graphically and numerically.