Approximation on Durrmeyer modification of generalized Szász-Mirakjan operators

This paper deals with the approximations of the functions by generalized Durrmeyer operators of Szasz-Mirakjan, which are linear positive operators. Several approximation results are presented well, and we estimate the approximation properties along with the order of approximation and the convergence theorem of the proposed operators. For an explicit explanation of the operators, we determine the properties using the weight function. A quantitative approach is discussed for the operators; quantitative Voronovskaya type and Gruss type theorems are established, showing the operators' more efficient work. We investigate the A$$ A $$-statistical convergence properties for the said operators, including the rate of approximation in a statistical sense. An important property for the rate of convergence of the operators is obtained in terms of the function with a derivative of the bounded variation. At last, the graphical representations and numerical analysis are discussed and shown to support our theoretical findings.