Approximation by Szasz-Mirakjan-Durrmeyer operators based on shape parameter λ

Abstract

In this work, we study several approximation properties of Sz ́asz- Mirakjan-Durrmeyer operators with shape parameter λ ∈ [−1, 1]. Firstly, we obtain some preliminaries results such as moments and central moments. Next, we estimate the order of convergence in terms of the usual modulus of continuity, for the functions belong to Lipschitz type class and Peetre’s K-functional, respectively. Also, we prove a Korovkin type approximation theorem on weighted spaces and derive a Voronovskaya type asymptotic the- orem for these operators. Finally, we show the comparison of the convergence of these newly defined operators to certain functions with some graphics and an error of approximation table.