A New Generalization of Szasz-Mirakjan Kantorovich Operators for Better Error Estimation

Abstract

In this paper, we construct a new sequence of Sz\'{a}sz-Mirakjan Kantorovich Operators $K_{n,\gamma}(f;x)$ depending on a parameter $\gamma$. We prove direct and local approximation properties of these operators. We obtain the operators $K_{n,\gamma}(f;x)$ to have better approximation results than classical Sz\'{a}sz-Mirakjan Kantorovich Operators for all $x\in[0,\infty)$, for any $\gamma>1$. Furthermore, we investigate the approximation results of these operators graphically and numerically. Moreover, we introduce new operators from $K_{n,\gamma}(f;x)$ that preserve affine functions and bivariate case of $K_{n,\gamma}(f;x)$. Then, we study their approximation properties and also illustrate the convergence of these new operators comparing with their classical cases.