Approximation by associated GBS operators of Sz$\acute{\text{a}}$sz-Mirakjan type operators

Abstract

In this article, the approximation properties of the Sz\'asz-Mirakjan type operators are studied for the function of two variables and the rate of convergence of the bivariate operators is determined in terms of total and partial modulus of continuity. An associated GBS (Generalized Boolean Sum)-form of the bivariate Sz\'asz-Mirakjan type operators is considered for the function of two variables to find an approximation of $B$-continuous and $B$-differentiable function in the B$\ddot{\text{o}}$gel's space. Further, the degree of approximation of the GBS type operators is found in terms of mixed modulus of smoothness and functions belonging to the Lipschitz class as well as a pioneering result is obtained in terms of Peetre $K$-functional. Finally, the rate of convergence of the bivariate Sz\'asz-Mirakjan type operators and the associated GBS type operators are examined through graphical representation for the finite and infinite sum which shows that the rate of convergence of the associated GBS type operators is better than the bivariate Sz\'asz-Mirakjan type operators and also a comparison is taken place for the bivariate operators with bivariate Kantorovich operators.