Computational error analysis of explicitly defined exponentially fitted Szász–Mirakyan operators

This manuscript presents a novel variant of the classical Szász-Mirakyan operators. The principal motivation is to obtain explicit closed-form expressions for the approximation error bounds, which are typically unavailable in general exponential preservation frameworks. In contrast to the standard operators, our proposed construction is specifically designed to reproduce the test functions $h_0(\chi )=1$ $h_{0}(\chi )=1$ and $h^{\ast }(\chi )=e^{2\chi }$ $h^{*}(\chi )=e^{2\chi }$ precisely. We examine the approximation characteristics of these operators within the space of continuous functions defined on compact intervals. A uniform convergence result is established through Korovkin-type theory, and the convergence rate is characterized via the classical modulus of continuity. Furthermore, a Voronovskaja-type asymptotic expansion is derived to elucidate the asymptotic behavior. We also investigate simultaneous approximation of derivatives alongside shape-preserving characteristics, particularly monotonicity and convexity preservation. Numerical experiments accompanied by graphical visualizations substantiate the theoretical findings, revealing that for the benchmark function $\phi (\chi )=e^{3\chi }$ $\varphi (\chi )=e^{3\chi }$ , the proposed operators achieve function-dependent improvement ratios: approximately 3 for $\phi (\chi )=e^{3\chi }$ $\varphi (\chi ) = e^{3\chi }$ and approximately 7 for $\psi (\chi )={\chi }^2{e}^{\chi }$ $\psi (\chi ) = \chi ^{2} e^{\chi }$ , in excellent agreement with the Voronovskaja-type asymptotic predictions.