关键词: 有理逼近, 有理插值, Newman结点, Newman型有理算子, Newman不等式, 逼近阶

Abstract: Rational approximation is an important and very vital topic in the theory of function approximation. In this paper, we study the approximation of the nonsmooth function $|x|$ by the Newman rational operator, by increasing $n$ nodes near the zero of the Newman constructed nodes. First, we introduce some main achievements on the rational interpolation to $|x|$. Then, by improving the Newman inequality, it improves from the original $e^{-\sqrt{n}}$ to $8e^{-2\sqrt{n}}$. From this, the approximation order of Newman-type rational operator approximating $|x|$ is $O(e^{-2\sqrt{n}})$, the result is better than the classical results of Newman.

Key words: rational approximation, rational interpolation, Newman nodes, Newman-type rational operator, Newman inequality, order of approximation