基于修正的第二类Chebyshev结点的Newman型有理插值函数对|x|的逼近

Recently Brutman and Passow considered Newman-type rational interpolation to ｜x｜ induced by arbitrary sets of symmetric nodes in [-1,1] and gave the general estimation of the approximation error.By their methods,one could establish the exact order of approximation for some special nodes.In the present note we consider the sets of interpolation nodes obtained by adjusting the Chebyshev roots of the second kind on the interval [0,1] and then extending this set to [-1,1] in a symmetric way.We show that in this case the exact order of approximation is O（ 1 n 2 ）.     

ON THE ORDER OF APPROXIMATION FOR THE RATIONAL INTERPOLATION TO |x|

The order of approximation for Newman-type rational interpolation to |x| is studied in this paper. For general set of nodes, the extremum of approximation error and the order of the best uniform approximation are estimated. The result illustrates the general quality of approximation in a different way. For thespecial case where the interpolation nodes are xi= (i/n)(i= 1,2,…,n;r>0) , it is proved that the exact order of approximation is O(1/n),O(1/nlogn) and O(1/n), respectively, corresponding to O1.     

On the order of approximation for the rational interpolation to |x|

The order of approximation for Newman-type rational interpolation to |x| is studied in this paper.For general set of nodes, the extremum of approximation error and the order of the best uniform approximation are estimated. The result illustrates the general quality of approximation in a different way. For the special case where the interpolation nodes are xi=(i/n)r(i=1,2, … ,n;r＞0), it is proved that the exact order of approximation is O(1/n), O(1/nlogn) and O(1/nr), respectively, corresponding to 0＜r＜1, r=1 and r＞1.     

THE DIVERGENCE OF LAGRANGE INTERPOLATION FOR ｜x｜^α（2〈α〈4）AT EQUIDISTANT NODES

It is a classical result of Bernstein that the sequence of Lagrange interpolation polumomials to ｜x｜ at equally spaced nodes in [-1, 1] diverges everywhere, except at zero and the end-points. In the present paper, toe prove that the sequence of Lagrange interpolation polynomials corresponding to ｜x｜^α （2 〈 α 〈 4） on equidistant nodes in [-1, 1] diverges everywhere, except at zero and the end-points.     

THE EXACT CONVERGENCE RATE AT ZERO OF LAGRANGE INTERPOLATION POLYNOMIAL TO ｜x｜^α

In this paper we present a generalized quantitative version of a result the exact convergence rate at zero of Lagrange interpolation polynomial to spaced nodes in [-1,1] due to M.Revers concerning f（x） = ｜x｜α with on equally     

The average errors for Hermite-Fejr interpolation on the Wiener space

For 1≤ p ∞, firstly we prove that for an arbitrary set of distinct nodes in [-1, 1], it is impossible that the errors of the Hermite-Fejr interpolation approximation in L p -norm are weakly equivalent to the corresponding errors of the best polynomial approximation for all continuous functions on [-1, 1]. Secondly, on the ground of probability theory, we discuss the p-average errors of Hermite-Fejr interpolation sequence based on the extended Chebyshev nodes of the second kind on the Wiener space. By our results we know that for 1≤ p ∞ and 2≤ q ∞, the p-average errors of Hermite-Fejr interpolation approximation sequence based on the extended Chebyshev nodes of the second kind are weakly equivalent to the p-average errors of the corresponding best polynomial approximation sequence for L q -norm approximation. In comparison with these results, we discuss the p-average errors of Hermite-Fejr interpolation approximation sequence based on the Chebyshev nodes of the second kind and the p-average errors of the well-known Bernstein polynomial approximation sequence on the Wiener space.     

THE DIVERGENCE OF LAGRANGE INTERPOLATION FOR ｜x｜^α

This paper shows that the sequence of Lagrange interpolation polynomials corresponding to the rune tion f（z） =｜x｜^α（1〈α〈2） on [-1,1] can diverge everywhere in the interval except at zero and the end-points.     

UNIFORM SUPERCONVERGENCE OF GALERKIN METHODS FOR SINGULARLY PERTURBED PROBLEMS

In this paper, we are concerned with uniform superconvergence of Galerkin methods for singularly perturbed reaction-diffusion problems by using two Shishkin-type meshes. Based on an estimate of the error between spline interpolation of the exact solution and its numerical approximation, an interpolation post-processing technique is applied to the original numerical solution. This results in approximation exhibit superconvergence which is uniform in the weighted energy norm. Numerical examples are presented to demonstrate the effectiveness of the interpolation post-processing technique and to verify the theoretical results obtained in this paper.     

SOME ARGUMENTS FOR RECOVERING THE FINITE ELEMENT~*

The dual argument is well known for recoving the optimal L_2-error of the finite element method in elliptic context. This argument, however, will lose efficacy in hyperbolic case. An expansion argument and an approximation argument are presented in this paper to recover the optimal L_2-error of finite element methods for hyperbolic problems. In particular, a second order error estimate in L_2-norm for the standard linear finite element method of hyperbolic problems is obtained if the exact solution is smooth and the finite element mesh is almost uniform, and some superconvergence estimates are also established for leas smooth solution.     

Optimization on class of operator equations in the probabilistic case setting

In this paper, we introduce a problem of the optimization of approximate solutions of operator equations in the probabilistic case setting, and prove a general result which connects the relation between the optimal approximation order of operator equations with the asymptotic order of the probabilistic width. Moreover, using this result, we determine the exact orders on the optimal approximate solutions of multivariate Preldholm integral equations of the second kind with the kernels belonging to the multivariate Sobolev class with the mixed derivative in the probabilistic case setting.     

ON LAGRANGE INTERPOLATION FOR [X|a (0＜a＜1)

We study the optimal order of approximation for |x|a (0 ＜ a ＜ 1) by Lagrange interpolation polynomials based on Chebyshev nodes of the first kind. It is proved that the Jackson order of approximation is attained.     

│x│的有理插值的若干注记

本文中我们构造了一个结点组,基于它定义的有理插值函数,对于任意给定的自然数k,对|x|的逼近能达到精确阶O(1/(n^klogn)).更重要的是,这样的构造揭示了一个本质:当结点向(|x|的唯一奇异点)零点集中时,|x|的有理插值逼近阶也随之更佳,这或许为将来本质性的自然结点组的构造提供了一种思路.     

On Lagrange interpolation for |<Emphasis Type="Italic">X</Emphasis>|<Superscript><Emphasis Type="Italic">α</Emphasis></Superscript> (0 < <Emphasis Type="Italic">α</Emphasis> < 1)

We study the optimal order of approximation for |x| ^α (0 < α < 1) by Lagrange interpolation polynomials based on Chebyshev nodes of the first kind. It is proved that the Jackson order of approximation is attained. Supported by the NSFC, 10601065.     

The average errors for lagrange interpolation on the Wiener space

For the weighted approximation in L _p -norm, we determine the asymptotic order for the paverage errors of Lagrange interpolation sequence based on the Chebyshev nodes on the Wiener space. We also determine its value in some special case.     

Lagrange插值和Hermite-Fejér插值在Wiener空间下的平均误差

在L_q-范数逼近的意义下,确定了基于Chebyshev多项式零点的Lagrange插值多项式列和Hermite-Fejér插值多项式列在Wiener空间下的p-平均误差的弱渐近阶.从我们的结果可以看出,当2≤q<∞,1≤p<∞时,基于第一类Chebyshev多项式零点的Lagrange插值多项式列和Hermite-Fejér插值多项式列的p-平均误差弱等价于相应的最佳逼近多项式列的p-平均误差.在信息基计算复杂性的意义下,如果可允许信息泛函为计算函数在固定点的值,那么当1≤p,q<∞时,基于第一类Chebyshev多项式零点的Lagrange插值多项式列和Hermite-Fejér插值多项式列在Wiener空间下的p-平均误差弱等价于相应的最小非自适应p-平均信息半径.     

ON THE ORDER OF APPROXIMATION FOR THE RATIONAL INTERPOLATION TO ｜x｜

The order of approximation for Newman-type rational interpolation to |x| is studied in this paper. For general set of nodes, the extremum of approximation error and the order of the best uniform approximation are estimated. The result illustrates the general quality of approximation in a different way. For the special case where the interpolation nodes are $x_i = \left( {\frac{i}{n}} \right)^r (i = 1,2, \cdots ,n;r > 0)$x_i = \left( {\frac{i}{n}} \right)^r (i = 1,2, \cdots ,n;r > 0) , it is proved that the exact order of approximation is O( \frac1n ),O( \frac1nlogn ) and O( \frac1n^r )O\left( {\frac{1}{n}} \right),O\left( {\frac{1}{{n\log n}}} \right) and O\left( {\frac{1}{{n^r }}} \right) , respectively, corresponding to 01.     

连续可导函数类的最优拉格朗日插值

在构造拉格朗日插值算法时,插值结点的选择是十分重要的.给定一个足够光滑的函数,如果结点选择的不好,当插值结点个数趋于无穷时,插值函数不收敛于函数本身.例如龙格现象:对于龙格函数f(x)=1/1+25x^2,如果拉格朗日插值的结点取[-1,1]上的等距结点,那么逼近的误差会随着结点个数增多而趋于无穷大⑴,由此可知插值结点的选择尤为重要.