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This paper presents a Stancu-type generalization of Szasz operators linked to Hermite polynomials. The convergence properties of these operators are discussed by utilizing Korovkin's theorem. Additionally, the paper explores approximation theorems for these operators using various tools, including Peetre's K-functional, classical and second-order modulus of continuity. The convergence rate for Lipschitz-type functions is estimated as well. Furthermore, the paper investigates the rate of weighted A-statistical convergence and demonstrates a Voronovskaya-type asymptotic result for these operators. [ABSTRACT FROM AUTHOR]

Runge's phenomenon reveals that interpolation methods lack uniform convergence of the sequence of the thus constructed polynomials to the function. The linear positive operators, however, can be relied on to ensure convergence across the whole domain. Further, linear positive operators can be modified suitably to approximate integrable functions. In this paper, we prove theoretical results on the estimates for the rate of approximation of a new sequence of operators. In addition, we find an error estimate for a larger class of functions, the functions of bounded variation. Finally, the theory is supported by suitable numerical examples, and the convergence estimates of the proposed operator are compared with the classical modified Bernstein–Durrmeyer operator. [ABSTRACT FROM AUTHOR]