On rational interpolation to |x| at the logarithmic nodes
ZHANG Huiming1， LI Jianjun2
(1.School of Mathematics and Physics， Hebei GEO University， Shijiazhuang 050031， China；2.Minzu College Affaliated to Hebei Normal University， Shijiazhuang 050091， China)
Abstract：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 Onlog n(1) 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 On2log n(1). 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|.
2024, Vol. 58 
