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

Abstract

In this article, we construct a new sequence of Szász-Mirakjan-Kantorovich operators denoted as $K_{n,\gamma}(f;x)$, which depending on a parameter $\gamma$. We prove direct and local approximation properties of $K_{n,\gamma}(f;x)$. We obtain that, if $\gamma>1$, then the operators $K_{n,\gamma}(f;x)$ provide better approximation results than classical case for all $x\in[0,\infty)$. Furthermore, we investigate the approximation results of $K_{n,\gamma}(f;x)$, 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 operators comparing with their classical cases.