Statistical convergence of Szasz-Mirakjan-Kantorovich-type operators and their bivariate extension

Abstract

Our main aim is to investigate the approximation properties for the summation integral type operators in statistical sense. In this regard, we prove the statistical convergence theorem using well known Korovkin theorem and the degree of approximation is determined. Also using weight function, the weighted statistical convergence theorem with the help of Korovkin theorem is obtained. The statistical rate of convergence in the terms of modulus of continuity and function belonging to the Lipschitz class are obtained. To support the convergence results of the proposed operators to the function, graphical representations take place and a comparison is shown with Szasz-Mirakjan-Kantorovich operators through examples. The last section deals with, a bivariate extension of the proposed operators to study the rate of convergence for the
function of two variables, additionally, convergence of the bivariate operators is shown with graphically.