On the order of approximation by modified summation-integral-type operators based on two parameters

Abstract In this article, we the study generalized family of positive linear operators based on two parameters, which are a broad many well-known operators, e.g., Baskakov-Durrmeyer, Baskakov-Szász, Szász-Beta, Lupa?-Beta, Lupa?-Szász, genuine Bernstein-Durrmeyer, and P?lt?nea. We first find direct estimates in terms second-order modulus continuity, then an estimate error Ditzian-Totik smoothness. Then discuss rate approximation for functions Lipschitz class. Furthermore, pointwise Grüss-Voronovskaja-type result also establish asymptotic formula quantitative form.