A Kind of Generalization of the Curve Type Node Configuration in R^S（S 〉 2）

A kind of generalization of the Curve Type Node Configuration is given in this paper,and it is called the generalized node configuration CTNCB in RS(S＞2).The related multivariate polynomial interpolation problem is discussed.It is proved that the CTNCB is an appropriate node configuration for the polynomial space PSn (S＞2).And the expressions of the multivariate Vandermonde determinants that are related to the Odd Curve Type Node Configuration in R2 are also obtained.     

Small perturbations of polynomial meshes

We show that the property of being a (weakly) admissible mesh for multivariate polynomials is preserved by small perturbations on real and complex Markov compacts. Applications are given to smooth transformations of polynomial meshes and to polynomial interpolation.     

一类多元全次数 Hermite 插值问题研究

In this paper we study a class of multivariate Hermite interpolation problem on 2~d nodes with dimension d ≥ 2 which can be seen as a generalization of two classical Hermite interpolation problems of d = 2. Two combinatorial identities are firstly given and then the regularity of the proposed interpolation problem is proved.     

On some aspects of multivariate polynomial interpolation

The purpose of this paper is to present some aspects of multivariate Hermite polynomial interpolation. We do not focus on algebraic considerations, combinatoric and geometric aspects, but on explicitation of formulas for uniform and non-uniform bivariate interpolation and some higher dimensional problems. The concepts of similar and equivalent interpolation schemes are introduced and some differential aspects related to them are also investigated. This revised version was published online in June 2006 with corrections to the Cover Date.     

Asymptotic Conditions for Degree Diminution Along Prescribed Lines

One of the problems in bivariate polynomial interpolation is the choice of a space of polynomials suitable for interpolating a given set of data. Depending on the number of data, a usual space is that of polynomials in 2 variables of total degree not greater than k. However, these spaces are not enough to cover many interpolation problems. Here, we are interested in spaces of polynomials of total degree not greater than k whose degree diminishes along some prescribed directions. These spaces arise naturally in some interpolation problems and we describe them in terms of polynomials satisfying some asymptotic interpolation conditions. This provides a general frame to the interpolation problems studied in some of our recent papers.     

ON HIGH DIMENSIONAL INTERPOLATION

1　IntroductionIn this paper,we letPn,… ,nsn′s( or Psn) be the ( s-variate) polynomial space of all real ( s-variate) polynomials with the degreeof each variate atmostn and use the usual multivariate notationwj =wj11 … wjss,| j| =j1 +… + js( j1 ,… ,js∈ Z+) .[1 ] and [2 ] have discussed the Cross Type Node Configuration ( CRTNC) and thecorresponding bivariate interpolation in R2 .In this paper,we considerRs={ ( w1 ,… ,ws) :wi ∈ R,i =1 ,… ,s} .We say that s( s-1 ) -dimensional h…     

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.     

Birkhoff型三角插值

1 引言 Birkhoff三角插值是近年来比较活跃的一个研究课题,涉及Birkhoff三角插值的研究文献也很多(如G.G.Lorentz~([1]),沈燮昌~([2])等综合性文章).     

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.     

Interpolation from spaces spanned by monomials

This is an extension and emendation of recent results on the use of Gauss elimination in multivariate polynomial interpolation and, in particular, ideal interpolation. Dedicated to Mariano Gasca on the occasion of his sixtieth birthday     

Polynomial interpolation in several variables

This is a survey of the main results on multivariate polynomial interpolation in the last twenty-five years, a period of time when the subject experienced its most rapid development. The problem is considered from two different points of view: the construction of data points which allow unique interpolation for given interpolation spaces as well as the converse. In addition, one section is devoted to error formulas and another to connections with computer algebra. An extensive list of references is also included. This revised version was published online in June 2006 with corrections to the Cover Date.     

Some extremal properties of multivariate polynomial splines in the metric Lp (Rd )

We constructed a kind of continuous multivariate spline operators as the approximation tools of the multivariate functions on the Bd instead of the usual multivariate cardinal interpolation oper-ators of splines, and obtained the approximation error by this kind of spline operators. Meantime, by the results, we also obtained that the spaces of multivariate polynomial splines are weakly asymptoti-cally optimal for the Kolmogorov widths and the linear widths of some anisotropic Sobolev classes of smooth functions on Bd in the metric Lp(Bd).     

On interpolation with products of positive definite functions

In this paper we consider the problem of scattered data interpolation for multivariate functions. In order to solve this problem, linear combinations of products of positive definite kernel functions are used. The theory of reproducing kernels is applied. In particular, it follows from this theory that the interpolating functions are solutions of some varational problems.     

Hermite插值算子在Orlicz空间内的逼近

插值算子逼近是逼近论中一个非常有趣的问题,尤其是以一些特殊的点为结点的插值算子的逼近问题很受人们的关注.研究了以第一类Chebyshev多项式零点为插值结点的Hermite插值算子在Orlicz范数下的逼近.     

关于球面上的一种Hermite插值

本文旨在提出单位球面上的一种Hermite插值格式.为此,本文首先研究了沿球面同轴圆周组上的Hermite插值问题,给出了三种适定的插值泛函组.然后研究了球面上的Hermite插值问题,给出了球面上Hermite插值的一种叠加插值法,即添加圆周组法.进一步将二者结合,导出了一类球面上Hermite插值的适定插值泛函组.为了说明这类适定插值泛函组的构造方法,在本文最后还给出了构造球面上低次Hermite适定插值泛函组的一些具体例子和数值算例.     

受限制多项式插值及在构造形函数空间中的应用

1引言 G.Strang指出：有限元法的新思想在于试控函数的选择，目标是选择这样的分片多项式，它们被少数几个方面的节点值确定，并仍具有我们需要的连续性和逼近度。受限制多项式插值空间就是这样一类空间，在P.G.Ciarlet~[4]的书中有较多的介绍，采用的方法是通过约束条件来决定试验空间，但正如[1]中指出的，这样约束条件欠直观，且容易产生一些不确     

多元分离子的构造与应用

本文研究了两类多元分离子的构造方法并提出计算公式.分析了多元分离子与Groebner 基的关系.把所求分离子应用到多元插值问题上,得到在字典序与广义字典序下用分离子表示的多元插值多项式.数值模拟显示了所述方法的有效性.     

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.     

修正的 Thiele-Werner型有理插值

Through adjusting the order of interpolation nodes, we gave a kind of modified Thiele-Werner rational interpolation. This interpolation method not only avoids the infinite value of inverse differences in constructing the Thiele continued fraction interpolation, but also simplifies the interpolating polynomial coefficients with constant coefficients in the Thiele-Werner rational interpolation. Unattainable points and determinantal expression for this interpolation are considered. As an extension, some bivariate analogy is also discussed and numerical examples are given to show the validness of this method.     

一种广义插值法

本文考虑一种广义插值问题，插值条件为小区间上的积分值，以弥补现有的插值方法在L2空间不再适用的不足，除了多项式插值外，还讨论了两种一次样条插值方法。