A new parametric formulation of Baskakov-Schurer-Szász operators with approximation properties

In this study, we introduce a novel parametric extension of Baskakov-Schurer-Sz\'asz operators, which extends the established classical version of these operators. We establish a Korovkin-type theorem for this class, along with a Gr"uss-Voronovskaya-type result, and analyze the rate of convergence. Beyond the standard convergence analysis, we extend our investigation into weighted function spaces, where the behavior of the parametric operators is examined in the presence of varying function weights. This aspect is particularly important for approximating functions with non-uniform behavior or those defined over unbounded domains. Moreover, we focus on the shape-preserving properties of the parametric operators. We prove that, under certain conditions, these operators preserve key geometric characteristics of the functions they approximate, such as monotonicity and convexity. This is a crucial feature in applications where the structural integrity of the original function must be maintained. Furthermore, we show that the classical Baskakov-Schurer-Szász operators emerge as a special case of our parametric generalization, thereby encompassing a broader range of operator classes and providing a more unified theoretical framework for approximation operators.