Approximation by Szász-Mirakyan operators preserving $e^{-2(p_{1}+p_{2})}$

Abstract

The present paper is dedicated to the modification of the bivariate generalized Sz\'{a}sz--Mirakyan operators while preserving the exponential functions $\exp(2,2)$ where $\exp(\tau_{1},\tau_{2})=e^{-\tau_{1} p_{1}-\tau_{2} p_{2}},\tau_{1},\tau_{2}\in\mathbb{R}_{0}^{+}$, and $p_{1},p_{2}\geq0$. We thoroughly investigate the weighted approximation properties and also obtain the convergence rate for these operators by utilizing a weighted modulus of continuity. Additionally, we delve into the order of approximation by investigating local approximation results through Peetre's $\mathcal{K}$-functional. Furthermore, we present the GBS (Generalized Boolean Sum) operators of Sz\'{a}sz--Mirakyan operators and obtain the order of approximation in terms of the Lipschitz class of B\"{o}gel continuous functions and the mixed modulus of smoothness. In order to enhance our theoretical findings and effectively showcase the efficiency of our developed operators, we have included a wide range of numerical and graphical examples using various values.