Problems Of Rational Interpolation Based On Different Nodes

There are many nonlinear problems to be solved in the fields of technical science and natural science.In general,the parametric form in the approximation problem can be presented in various forms and can contain recessive parameters.Since the nonlinear problem is very common,it is difficult to boil down to the general theory.As an important special form of nonlinear approximation,rational approximation has attracted people’s attention in practice and applications.The problems of rational approximation to the non-smooth function |x| has been paid more and more attention,some from the expanding scope of approximation of |x| function in x,the other is starting from a set of corresponding nodes,but for the general case,which is less on rational approximation to |x|^α,through the use of the predecessors’ research methods and ideas,this article will be to function |X|α,α∈[1,2),from a set of rational interpolation nodes caused by different distribution and structural characteristics of the different strength of approximation degree of strength for further discussion.In this paper,we study the approximation problem of non-smooth function |x|^α on[-1,1].By constructing four different types of nodes,furthermore,we discuss the approximation order of rational operator rn（x;x）to |x|^α respectively.The main contents are as follows:In the first part of the article,the background,the development process and the main research status of the rational approximation are introduced.The second part discusses the node of a special group-the second kind of adjustment of the Chebyshev nodes group,this kind of node set of symmetrical about 0.5,and is dense near both 0 and 1,while sparse near point 0.5,that is focused on both ends of the nodes in the interval[0,1],and spread in the middle,analysis proves that the exact approximation order for this case is O(1/n^2α).The third part mainly shows that by increasing the number of nodes in the vicinity of zero,that is,by increasing the number of nodes to increase the approximation order of r_n（X;x）to|x|^α,let X= {X_k=Sinkπ/2n²}_k=1^n-1∪{x_k=sinkπ/2n}_k=1^n-1,we get this byincreasing the number of nodes,and the results show that the method of raising the approximationorder is feasible and the final approximation order is O(1/n^2αlogn).In the fourth part,the generalasymptotic formula of the convergence rate of Newman-α type rational operator approximation is obtained by choosing a new set of nodes,let X={±q^k}_k=1^n-1,where q = e-αn,and αn satisfies certain conditions,the general result is given.In the fifth part,we discuss the case of Newman node group similar to the exponent of power,which mainly gives the case of two kinds of node groups.These two types of nodes are special to the general method,the final summary of the general form of the results.The sixth part summarizes the whole paper and looks forward to the future research prospects.