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

Abstract

Our main aim is to investigate the approximation properties of the summation integral type operators in a 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 the weight function, the weighted statistical convergence theorem with the help of the Korovkin theorem is obtained. The statistical rate of convergence in the terms of modulus of continuity and function belonging to the Lipschitz class is determined. To support the convergence results of the proposed operators to the function, graphical representations take place and a comparison is shown with Sz?sz-Mirakjan-Kantorovich operators through examples. The last section deals with, a bivariate extension of the proposed operators to determine the approximation of the function of two variables, additionally, the rate of convergence is estimated as well as the convergence of the bivariate operators is shown by graphical representations.