Approximation properties of Gamma operators

In this paper the approximation properties of Gamma operators Gn 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 Gn for sign function. And then, this asymptotic formula combining with a metric form Ωx(f,λ) is used to derive an asymptotic estimate on the rate of convergence of Gamma operators Gn 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 Gn(|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 Gn for the absolutely continuous functions.