| Function approximation is an important part of the function approximation problem,we often encounter such kind of problems in the research fields of mathematics and practical applications,i.e.,looking for a function in a selected class of functions,which turns out to be the approximate representation of the given function.Also,we could calculate the error between these two functions.The linear approximation of a function,with all the parameters to be determined appear in a linear form,is aided by the approximation of a linear combination of the given function.The approximation problem of the typical non-smooth function f(x)=|x| has received more and more attention.Scholars did a lot of research on the linear(polynomial)approximation.However,the establishment of the calculation method as well as the error analysis is more difficult than that of the polynomial's.Therefore,researchers pay more attention to the rational approximation.In 1964,Newman first proved that the best rational approximation Rn(|x|)of |x| on[-1,1]was much better than the polynomial approximation.In recent years,more conclusions emerged.Some scholars began to study the expansion of the domain of |x|,some discussed the corresponding node sets,and they had different conclusions.Nevertheless,the construction of node sets of rational interpolation to |x|,distribution and degree of convergence between different approximation characteristics and rational relations remain many problems to be solved and the study of these problems has important significance for rational approximation theory.Newman type interpolation is a kind of rational interpolation that we are familiar with,the rational operator rn,a(X;x)=x?pn(x)-pn(-x)/pn(x)+pn(-x),which(?)has the advantages of simplestructure,good approximation effect,and its calculation is also simple and convenient.In the current literature,many researchers focus on the rational approximation to |x| with different interpolation nodes in the interval[-1,1],and there is not much research on general situation of|x|?.Therefore,the follow-up study can study the rational approximation to |x|?(1 ??<2)at the known node sets on one hand,and on the other hand,it is also possible to study the construction of different node sets and rational approximation of |x|?(1 ? ?<2),on the basis of which,it is also possible to study the rational approximation of |x|? in infinite or semi-axis intervals.This article is divided into four chapters:In the first chapter,we firstly introduce the research purpose and significance of the topics discussed in this article,then we give a brief description of the research status and development trends at home and abroad,and finally estabish the main results of this paper.In the second chapter,we introduce the convergence rate of Newman-? type rational operators to |x|?(1??<2).When the |x| is highly dense in 0 on the set of nodes x1=1/m2,x2=2/m2,…,Xm-1=(m-1)/m2,Xm=1/m,xm+1= 2/m,…,X2m-2 =(m?-1)/m,X2m-1 = 1.the exact order of approximation is O(1/n2?logn).On the basis of the above,the third chapter introduces the construction of Newman-? type rational operator rn,?(X;x)in the[-1,1]interval approaching |x|?(1??<2),X={xi=(i/n)r,i=1,2,…,n,r>0} is the constructed node sets,meanwhile the order of approximation is obtained,which are O(1/n?)?O(1/n?logn)?O(1/n?r),corresponding to the case0<r<1?r=1?r>1,respectively.In the fourth chapter,on the basis of the rational interpolation of the former study |x| in the tangent node group,when 1??<2,we prove the approximation order of Newman-? type rationaloperators to |x|?,and the exact approximation order O(1/n?logn)is obtained with the nodegroup X,the tangent point set {tank?/4n}k=1n.The results in this paper generalize the interpolation result of the former,and also include the approximation result of the former on the research problem of rational interpolation. |
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