Statistical convergence of bivariate generalized Bernstein operators via four-dimensional infinite matrices

Abstract

Our main aim in this work is to construct an original extension of bivariate Bernstein type operators based on multiple shape parameters to give an application of four-dimensional infinite matrices to approximation theory, and prove some Korovkin theorems using two summability methods: a statistical convergence method which is stronger than the classical case and a power series method. We obtain the rate of generalized statistical convergence, and the rate of convergence for the power series method. Moreover, we provide some computer graphics to numerically analyze the efficiency and accuracy of convergence of our operators and obtain corresponding error plots. All the results that have been obtained in the present paper can be extended to the case of $n$-variate functions.