On rational interpolation to|x|at the logarithmic nodes
The rational approximation of|x|is a very important topic in approximation theory.Firstly,the rational interpolation of|x|at a new class of node groups(logarithmic nodes)is studied in this paper,and it is obtained that the exact order of approximation is O(1/nlog n)by using appropriate scaling methods for approximation errors of|x|.Then,by adding some nodes with the same structure near the zero point,the approximation order can be increased to O(1/n2log n).Finally,the structure of five node groups with the same approximation order is analyzed to reveal their essence:because four types of node groups are equivalent to the logarithmic node groups,the error of|x|in five types of node groups is the same order.This conclusion indicates that the structural characteristics of node groups play a crucial role in the rational interpolation problem of|x|.
the logarithmic nodesrational interpolationNewman-type rational operatorsorder of approximation
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