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.

In this paper, by introducing the concept of neighbor points and using the first-order difference of control points of Catmull-Clark surfaces, we obtain the rate of convergence of control meshes of Catmull-Clark surface. By the result of convergence we derive a computational formula of subdivision depth for Catmull-Clark surfaces.

Asymptotic behavior of two Bernstein-type operators is studied in this paper. In the first case, the rate of convergence of a Bernstein operator for a bounded function f is studied at points x where f (x+) and f (x-) exist. In the second case, the rate of convergence of a Szasz operator for a function f whose derivative is of bounded variation is studied at points X where f (x+) and f (x-) exist. Estimates of the rate of convergence are obtained for both cases and the estimates are the best possible for continuous points.

In this paper we estimate the rate of convergence of the Durrmeyer-Bezier operators Dn,α: (f, x) for functions of bounded variation and prove that the Dn, α: (f, x) converge to the limit 1/(α+1) f (x+)+α/(α+1)f(x-) for functions of bounded variation f (t). Our result improves and extends the result of S. Guo