Rational Approximation Of A Class Of Nonlinear Functions

Function approximation theory is an important part of function theory,and the main research content is the approximate representation of a function.No matter in the field of mathematics research or practical application,it has an irreplaceable position.The problem of function approximation is to select a kind of simple function p(x)to replace the known function f(x),and to find the error caused by the selected simple function to approximate the known function.Rational function approximation,as a vital research topic in function approximation theory,naturally attracts scholars' attention,especially for the rational approximation of the nonsmooth function |x|.At the end of the 19th century,Chebyshev and de la Vallee Poussin began to study the best approximation of rational functions on the whole real axis.The problem of approximation to nonsmooth functions was first proposed by Bernstein in 1913.In 1964,Newman constructed the rational functions rn(x)to approximate |x|,then it was found that the approximation effect is much better than the polynomial approximation.After that,the approximation problem of |x| has attracted the attention of a large number of scholars,and they began to study and improve its approximation problem by expanding to different intervals and different nodes.This paper is divided into four chapters:the first chapter mainly introduces the research purpose and significance of the problems discussed in this paper,then briefly describes the research status and development trend at home and abroad,and finally gives the main work of this paper.In the second chapter,we mainly discuss the construction of a Newman-? type rational operator,by utilizing it to approximate a class of non-smooth functions,the convergence rate is studied.It is proved that the order of approximation is O(1/(n3? logn))when the modified Chebyshev nodes are selected as node group X,and it is verified that the result is optimal under such a construction.In essence,subdivision nodes can be further constructed and in that case,the order of approximation is O(1/(n(k+1)? log n)).Chapter three mainly discusses the node group based on a node group X=?xk=log((n+k)/n)}k=1n,the Newman type rational operator is constructed.The convergence rate of approximation to a class of non-smooth functions is discussed,which is O(1/(n log(n)))regarding X.Moreover,if the operator is constructed based on further subdivision nodes,the convergence rate is O(1/(n2 log(n))).The result in this paper is superior to the approximation results based on equidistant nodes,Chebyshev nodes of the first kind,and Chebyshev nodes of the second kind.The fourth chapter mainly discusses further research on the rational approximation of logarithmic nodes based on the third chapter.Based on node group X={xk=1/n log((n+k)/n)}k=1n,,we construct Newman-? type rational operator to approximate the convergence rate of a class of nonsmooth functions,and obtain the convergence rate O(1/(n2? log(n))).In addition,considering the relationship between the construction property of nodes and the approximation order,the operator can be constructed on the basis of further subdivision of nodes,the convergence rate is O(1/(n(k+1)? log(n))).The fifth chapter is the summarize of the whole paper.The problems that have not been solved in this paper and the areas that need to be improved are analyzed in this chapter.Furthermore,some future research prospects are proposed.