Approximation by bivariate Szász–Mirakyan operators preserving e^-2 (p₁ + p₂)

The present paper is dedicated to the modification of the bivariate generalized Sz? sz? Mirakyan operators while preserving the exponential functions (2, 2) where (₁, ₂) = e^-₁ p₁ - ₂ p₂, ₁, ₂ R₀^+, and p₁, p₂ 0. 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 K -functional. Furthermore, we present the GBS (Generalized Boolean Sum) operators of Sz? sz? Mirakyan operators and obtain the order of approximation in terms of the Lipschitz class of B? 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 examples using various values.