Abstract: Since Newman's rational operator has a good approximation for $|x|$, we consider the approximation of $|x|^{\alpha}$ by a Newman-$\alpha$ rational operator. In this paper, we discuss the convergence rate of the operator Newman-$\alpha$ at the adjusted tangent nodes $X=\{\tan^{2}\frac{k\pi}{4n}\}_{k=1}^{n}$, and finally obtain the exact approximation order $O(\frac{1}{n^{2\alpha}})$. The result not only contains the approximation result in the case of $\alpha=1$, but it is better than the conclusion when the node group is selected for the first and the second type of Chebyshev nodes, equidistant nodes etc.

Key words: rational approximation, Newman-$\alpha$ type rational operator, approximation order