New exponential operators connected with \(a^2+x^2\): a generalization of Post-Widder and Ismail-May operators

The present study offers a general exponential operator connected with \(a^2+x^2\) $a^2+x^2$ , for positive real a. We estimate the asymptotic formula for simultaneous and ordinary approximation of the constructed operator. We also consider the limiting case \(a \rightarrow 0\) $a\to 0$ , which provides Post-Widder operators. We graphically present the convergence of the operator to two functions, viz., “ \(x \sin (x)\) $xsin(x)$ ” and “ \(-\frac{x}{2} \cos (\pi x)\) $-\frac{x}{2}cos(\pi x)$ ”. In addition, we analyze each particular case of the defined operator and determine the optimal value of a, that would yield the greatest approximation. This allows us to compare the well-known operators in the literature, especially the Post-Widder operators and the Ismail-May operators. In the end, we conclude that the convergence of the operators improves as the value of a approaches zero from the right side.