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 polynomials to |x| at equally spaced nodes in [-1, 1] diverges everywhere, except at zero and the end-points. In the present paper, we 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 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 Divergence of Lagrange Interpolation for |x| at Equidistant Nodes

In 1918 S. N. Bernstein published the surprising result that the sequence of Lagrange interpolation polynomials to |x| at equally spaced nodes in [−1, 1] diverges everywhere, except at zero and the end-points. In the present paper, we prove that the sequence of Lagrange interpolation polynomials corresponding to |x|^α (0<α1) on equidistant nodes in [−1, 1] diverges everywhere in the interval except at zero and the end-points.     

THE DIVERGENCE OF LAGRANGE INTERPOLATION FOR |x|~a

This paper shows that the sequence of Lagrange interpolation polynomials corresponding to the function f(x) =|x|~a(1相似文献   

THE DIVERGENCE OF LAGRANGE INTERPOLATION FOR |x|α

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

一个无限可微函数类的最优Lagrange 插值

本文研究\,$[-1,1]$上的一个无限可微函数类$F_\infty$在空间$L_\infty[-1,1]$及加权空间$L_{p,\omega}[-1,1]$, $1\le p< \infty$ ($\omega$是$(-1,1)$上的非负连续可积函数)的最优Lagrange插值.我们证明了基于首项系数为1且于$L_{p,\omega}[-1,1]$上有最小范数的多项式零点的Lagrange插值对$1\le p< \infty$是最优的. 同时我们给出了当结点组包含端点时的最优结点组.     

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.     

On Lagrange interpolatory parabolas to $\left | x\right | ^{\alpha }$ at equally spaced nodes

In this paper we present a generalized quantitative version of a result due to D. L. Berman concerning the exact convergence rate at zero of Lagrange interpolation polynomials to | x| ^a\left | x\right | ^{\alpha } based on equally spaced nodes in [-1, 1]. The estimates obtained turn out to be best possible.     

On Divergence of Hermite—Fejér Interpolation to f(z)=z in the Complex Plane

We show that for a broad class of interpolatory matrices on [-1,1] the sequence of polynomials induced by Hermite—Fejér interpolation to f(z)=z diverges everywhere in the complex plane outside the interval of interpolation [-1,1] . This result is in striking contrast to the behavior of the Lagrange interpolating polynomials. June 15, 1998. Date accepted: January 26, 1999.     

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.     

On the Zero-Divergence of Equidistant Lagrange Interpolation

 In 1942, P. Szász published the surprising result that if a function f is of bounded variation on [−1, 1] and continuous at 0 then the sequence of the equidistant Lagrange interpolation polynomials converges at 0 to . In the present note we give a construction of a function continuous on [−1, 1] whose Lagrange polynomials diverge at 0. Moreover, we show that the rate of divergence attains almost the maximal possible rate.     

On the Zero-Divergence of Equidistant Lagrange Interpolation

 In 1942, P. Szász published the surprising result that if a function f is of bounded variation on [−1, 1] and continuous at 0 then the sequence of the equidistant Lagrange interpolation polynomials converges at 0 to . In the present note we give a construction of a function continuous on [−1, 1] whose Lagrange polynomials diverge at 0. Moreover, we show that the rate of divergence attains almost the maximal possible rate. (Received 2 February 2000)     

改进Lagrange插值多项式误差上界的系数

当用Lagrange插值多项式逼近函数时,重要的是要了解误差项的性态.本文研究具有等距节点的Lagrange插值多项式,估计了Lagrange插值多项式逼近函数误差项的上界,改进了小于5次Lagrange插值多项式逼近函数误差界的系数.     

GLOBAL SMOOTHNESS PRESERVATION BY BIVARIATE INTERPOLATION OPERATORS

1　IntroductionIn the case when Pn(f,x) represents the univariate interpolation polynomial of Her-mite-Fejér based on Chebyshev nodesof the firstkind or the univariate interpolation polyno-mials of Lagrange based on Chebyshev nodes of the second kind and± 1 ,or the univariaterational Shepard operators,the following result of partial preservation of global smoothnessis proved in[4] :If f∈Lip M(α;[-1 ,1 ] ) ,0 <α≤ 1 ,then there existsβ=β(α) <α and M′>such thatω(Pn(f ) ;h)≤ M′h…     

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 due to M. Revers concerning the exact convergence rate at zero of Lagrange interpolation polynomial to f(x) = |x|α with on equally spaced nodes in [-1, 1].     

Lagrange基函数的复矩阵有理插值及连分式插值

1引言 矩阵有理插值问题与系统线性理论中的模型简化问题和部分实现问题有着紧密的联系~[1][2]，在矩阵外推方法中也常常涉及线性或有理矩阵插值问题~[3]。按照文~[1]的阐述。目前已经研究的矩阵有理插值问题包括矩阵幂级数和Newton-Pade逼近。Hade逼近，联立Pade逼近，M-Pade逼近，多点Pade逼近等。显然，上述各种形式的矩阵Pade逼上梁山近是矩     

Lagrange Interpolation for the Disk Algebra: The Worst Case

We consider Lagrange interpolation polynomials for functions in the disk algebra with nodes on the boundary of the unit disk. In case that the closure of the set of nodes does not cover the boundary of the unit disk we prove that there exists a residual set of functions in the disk algebra, such that the Lagrange interpolation polynomials of each of these functions form a dense subset of the space of all holomorphic functions defined on the unit disk.     

On One class of Hermite projectors

We conjecture that every ideal projector on \({\mathbb {C}}\left[ x_1,\ldots ,x_d\right] \) whose kernel is generated by precisely d polynomials is Hermite (i.e., the limit of Lagrange interpolation projectors). We validate this conjecture in case the d generators of the kernel have no roots at infinity.     

Divergence of interpolation processes on sets of the second category

C([0, 1]) is the space of real continuous functions f(x) on [0, 1] and ω(δ) is a majorant of the modulus of continuity ω(f, δ), satisfying the condition \(\mathop {\overline {\lim } }\limits_{n \to \infty } \omega (1/n) \ln n = \infty \) . A solution is given to a problem of S. B. Stechkin: for any matrix \(\mathfrak{M}\) of interpolation points there exists an f(x) ? c([0, 1]), ω (f, δ) = o{ω(δ)} whose Lagrange interpolation process diverges on a set ? of second category on [0, 1].     

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.