Multivariate Interpolation Using Carlitz's Inversion Formulas

Multivariate rational exponential Lagrange interpolation formulas, Hermite interpolation formulas, and Hermite–Fejér interpolation formulas of the Newton type are established by using Carlitz's inversion formulas. The recurrence relation for constructing Lagrange interpolation is also given. In addition, by setting q1 in the obtained formulas, we obtain the corresponding polynomial interpolation formulas with combinatorial form.     

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.     

关于球面上的Lagrange插值

1引 言 单位球面上的插值问题一直是三元插值问题中比较受关注的部分.近年来,球面上的 Lagrange插值问题已经得到了很好地解决.例如[1]中给出了构造单位球面上的Lagrange 插值适定结点组的一种方法：添加圆周法.[2]和[3]中研究了单位球面上的多项式插值问题,给出了构造单位球面上的插值适定结点组的另外两种方法.     

On a Hermite interpolation on the sphere

The purpose of this paper is to put forward a kind of Hermite interpolation scheme on the unit sphere. We prove the superposition interpolation process for Hermite interpolation on the sphere and give some examples of interpolation schemes. The numerical examples shows that this method for Hermite interpolation on the sphere is feasible. And this paper can be regarded as an extension and a development of Lagrange interpolation on the sphere since it includes Lagrange interpolation as a particular case.     

多元分次Lagrange插值

对多元多项式分次插值适定结点组的构造理论进行了深入的研究与探讨.在沿无重复分量代数曲线进行Lagrange插值的基础上,给出了沿无重复分量分次代数曲线进行分次Lagrane插值的方法,并利用这一结果进一步给出了在R~2上构造分次Lagrange插值适定结点组的基本方法.另外,利用弱Gr(o|¨)bner基这一新的数学概念,以及构造平面代数曲线上插值适定结点组的理论,进一步给出了构造平面分次代数曲线上分次插值适定结点组的方法,从而基本上弄清了多元分次Lagrange插值适定结点组的几何结构和基本特征.     

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.     

Intertwining unisolvent arrays for multivariate Lagrange interpolation

Generalizing a classical idea of Biermann, we study a way of constructing a unisolvent array for Lagrange interpolation in C^n+m out of two suitably ordered unisolvent arrays respectively in Cⁿ and C^m. For this new array, important objects of Lagrange interpolation theory (fundamental Lagrange polynomials, Newton polynomials, divided difference operator, vandermondian, etc.) are computed. AMS subject classification 41A05, 41A63     

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.     

Lagrange interpolation and finite element superconvergence

We consider the finite element approximation of the Laplacian operator with the homogeneous Dirichlet boundary condition, and study the corresponding Lagrange interpolation in the context of finite element superconvergence. For d‐dimensional Q_k‐type elements with d ≥ 1 and k ≥ 1, we prove that the interpolation points must be the Lobatto points if the Lagrange interpolation and the finite element solution are superclose in H¹ norm. For d‐dimensional P_k‐type elements, we consider the standard Lagrange interpolation—the Lagrange interpolation with interpolation points being the principle lattice points of simplicial elements. We prove for d ≥ 2 and k ≥ d + 1 that such interpolation and the finite element solution are not superclose in both H¹ and L² norms and that not all such interpolation points are superconvergence points for the finite element approximation. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 33–59, 2004.     

On the Negative Extremums of Fundamental Functions of Lagrange Interpolation Based on Chebyshev Nodes

In this paper, we investigate the negative extremums of fundamental functions of Lagrange interpolation based on Chebyshev nodes. Moreover, we establish some companion results to the theorem of J. Szabados on the positive extremum.     

Lagrange Interpolation on a Sphere

§ 1.Ｉｎｔｒｏｄｕｃｔｉｏｎ　ＬｅｔｎｂｅａｎｏｎｎｅｇａｔｉｖｅｉｎｔｅｇｅｒａｎｄＳ ={(ｘ ,ｙ ,ｚ)∈Ｒ3 |ｘ2 + ｙ2 +ｚ2 =1 }ｂｅｔｈｅｕｎｉｔｓｐｈｅｒｅｉｎＲ3 .Ｐ( 2 )ｎ ａｎｄＰ( 3 )ｎ ｄｅｎｏｔｅｔｈｅｓｐａｃｅｏｆａｌｌｂｉｖａｒｉａｔｅｐｏｌｙｎｏｍｉａｌｓｏｆｔｏｔａｌｄｅｇｒｅｅ≤ｎａｎｄｔｈｅｓｐａｃｅｏｆａｌｌｔｒｉｖａｒｉａｔｅｐｏｌｙｎｏｍｉａｌｓｏｆｔｏｔａｌｄｅｇｒｅｅ≤ｎｒｅｓｐｅｃｔｉｖｅｌｙ ,ｉ.ｅ .Ｐ( 2 )ｎ =∑0≤ｉ+ｊ≤ｎａｉｊｘｉｙｊ｜ａｉｊ ∈Ｒ ,Ｐ( 3 …     

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     

球面上的Lagrange插值

In this paper, we obtain a properly posed set of nodes for interpolation on a sphere. Moreover it is applied to construct properly posed set of nodes for Lagrange interpolation on the trivariate polynomial space of total degree n.     

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 numerical stability of barycentric Lagrange interpolation

The Lagrange representation of the interpolating polynomialcan be rewritten in two more computationally attractive forms:a modified Lagrange form and a barycentric form. We give anerror analysis of the evaluation of the interpolating polynomialusing these two forms. The modified Lagrange formula is shownto be backward stable. The barycentric formula has a less favourableerror analysis, but is forward stable for any set of interpolatingpoints with a small Lebesgue constant. Therefore the barycentricformula can be significantly less accurate than the modifiedLagrange formula only for a poor choice of interpolating points.This analysis provides further weight to the argument of Berrutand Trefethen that barycentric Lagrange interpolation shouldbe the polynomial interpolation method of choice.     

样条型矩阵有理插值

The matrix valued rational interpolation is very useful in the partial realization problem and model reduction for all the linear system theory. Lagrange basic functions have been used in matrix valued rational interpolation. In this paper, according to the property of cardinal spline interpolation, we constructed a kind of spline type matrix valued rational interpolation, which based on cardinal spline. This spline type interpolation can avoid instability of high order polynomial interpolation and we obtained a useful formula.     

Interp olation by Bivariate Polynomials Based on Multivariate F-truncated Powers

The solvability of the interpolation by bivariate polynomials based on multivariate F-truncated powers is considered in this short note. It unifies the pointwise Lagrange interpolation by bivariate polynomials and the interpolation by bivariate polynomials based on linear integrals over segments in some sense.     

A Short Proof of Krattenthaler Formulas

With an effort to investigate a unified approach to the Lagrange inverse Krattenthaler established operator method we finally found a general pair of inverse relations, called the Krattenthaler forumlas. The present paper presents a very short proof of this formula via Lagrange interpolation. Further, our method of proof declares that the Krattenthaler result is unique in the light of Lagrange interpolation. Received April 9, 1999, Accepted December 7, 2000     

A Note on a Theorem of J. Szabados

In this note, we establish a companion result to the theorem of J. Szabados on the maximum of fundamental functions of Lagrange interpolation based on Chebyshev nodes.     

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.