Weighted Approximation by Szász–Mirakyan–Durrmeyer Operators Reproducing Exponential Functions

We examine a Szász–Mirakyan–Durrmeyer type operator that reproduces the functions 1 and e2ax for a fixed parameter a>0. While its exponential reproduction property has been described in the classical literature, the effect of exponential weights on its approximation behavior has not been studied. In this work, we provide a detailed analysis of the operator in weighted spaces and show that combining exponential reproduction with weighted norms improves the approximation behavior for exponentially growing functions. We also prove that the corresponding sequence of operator norms remains uniformly bounded for a family of exponential weights, ensuring the stability of the operators in the weighted framework. Moreover, we establish new Korovkin-type approximation theorems involving weighted convergence and obtain sharp uniform error estimates in the presence of exponential weights. These results extend the classical theory to weighted exponential settings and highlight several quantitative features that do not arise in the classical case.