Approximation properties of Gamma operators

Abstract In this paper the approximation properties of Gamma operators G n are studied to the locally bounded functions and the absolutely continuous functions, respectively. Firstly, in Section 2 of the paper a quantitative form of the central limit theorem in probability theory is used to derive an asymptotic formula on approximation of Gamma operators G n for sign function. And then, this asymptotic formula combining with a metric form Ω x ( f , λ ) is used to derive an asymptotic estimate on the rate of convergence of Gamma operators G n for the locally bounded functions. Next, in Section 3 of the paper the optimal estimate on the first order absolute moment of the Gamma operators G n ( | t − x | , x ) is obtained by direct computations. And then, this estimate and Bojanic–Khan–Cheng's method combining with analysis techniques are used to derive an asymptotically optimal estimate on the rate of convergence of Gamma operators G n for the absolutely continuous functions.