基于修正的第二类Chebyshev结点的Newman型有理插值函数对|x|的逼近

On Newman-Type Rational Interpolation to |x| at the Adjusted Chebyshev Nodes of the Second Kind

中文关键词:  Newman型有理插值函数  修正的第二类Chebyshev结点  逼近阶.

英文关键词:Newman-type rational interpolation  adjusting the Chebyshev roots of the second kind  exact order of approximation.

中文摘要:

      最近Brutman和Passow给出了基于$[-1,1]$上任意一组对称结点的Newman型有理插值函数逼近$|x|$的一般结论。运用他们的方法，可以得出某些特殊结点上的精确逼近阶。本文，讨论了修正的第二类Chebshev结点上的情况，得出此时精确的逼近阶为$O\left(\frac{1}{n^2}\right)$。

英文摘要:

      Recently Brutman and Passow considered Newman-type rational interpolation to $|x|$ induced by arbitrary sets of symmetric nodes in $[-1,1]$ and gave the general estimation of the approximation error. By their methods, one could establish the exact order of approximation for some special nodes. In the present note we consider the sets of interpolation nodes obtained by adjusting the Chebyshev roots of the second kind on the interval $[0,1]$ and then extending this set to $[-1,1]$ in a symmetric way. We show that in this case the exact order of approximation is $O(\frac{1}{n^2})$.