Approximation for bounded functions and absolutely continuous functions by Beta operators
In this paper, the pointwise approximation properties of Beta operators (B) over bar (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 (B) over tilde (n) for the bounded functions. Next, we give the optimal estimate on the first-order absolute moment of the Beta operators B-n(vertical bar t - x vertical bar, x) by direct computations. Then, we use this estimate and Bojanic-Cheng-Khan's method combining with some analysis techniques to derive an asymptotically optimal estimate on the rate of convergence of Beta operators (B) over bar (n) for the absolutely continuous functions. (c) 2006 Elsevier Ltd. All rights reserved.
absolutely continuous functions beta operators approximation property probabilistic methods