Approximation of associated GBS operators by Szasz-Mirakjan type operators

TL;DR

This paper investigates the approximation capabilities of bivariate Szasz-Mirakjan type operators and their associated GBS operators, demonstrating improved convergence rates through theoretical analysis and graphical comparisons.

Contribution

It introduces and analyzes the approximation properties of GBS form of bivariate Szasz-Mirakjan operators, providing new convergence rate results and graphical validation.

Findings

GBS operators have better convergence than original operators.

Approximation degree is characterized using modulus of smoothness.

Graphical results confirm theoretical improvements.

Abstract

In this article, the approximation properties of the Szasz-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 Szasz-Mirakjan type operators are considered for the function of two variables to find an approximation of B-continuous and B-differentiable function in the Bogel'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 Szasz-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 Szasz-Mirakjan type operators.