Approximation for bounded functions and absolutely continuous functions by beta operators

In this paper, the pointwise approximation properties of Beta operators n are studied to the bounded functions and the absolutely continuous functions, respectively. First, we use the asymptotic form of the central limit theorem in probability theory to derive an asymptotic estimate on the rate of convergence of Beta operators n for the bounded functions. Next, we give the optimal estimate on the first-order absolute moment of the Beta operators Bn(t−χ,χ) by direct computations. Then, we use this estimate and Bojanic-Cheng-Khanʹs method combining with some analysis techniques to derive an asymptotically optimal estimate on the rate of convergence of Beta operators n, for the absolutely continuous functions.