插值问题

First, we introduce and discuss the various methods of multivariate polynomial interpolation in the literature. Based on this study, we state multivariate Lagrange interpolation over again from algebraic geometry viewpoint:Given different interpolation nodes A1,A2 .....,An in the affine n-dimensional space Kn, and accordingly function values fi(i = 1,..., m), the question is how to find a polynomial p K[x1, x2,...,xn] satisfying the interpolation conditions:where X=(x1,X2,....,xn). Similarly with univariate problem, we have provedTheorem If the monomial ordering is given, a minimal ordering polynomial satisfying conditions (1) is uniquely exsisted.Such a polynomial can be computed by the Lagrange-Hermite interpolation algorithm introduced in chapter 2. Another statement for Lagrange interpolation problem is:Given monomials 1 ,2 ,.....,m from low degree to high one with respect to the ordering, some arbitrary values fi(i= 1,..., m), find a polynomial p, such thatIf there uniquely exists such an interpolation polynomial p{X, the interpolation problem is called properly posed.

文中首先对现有的多元多项式插值方法作了一个介绍和评述，在此基础上我们从代数几何观点重新讨论了多元Lagrange插值问题：给定n维仿射空间K~n中两两互异的点A_1，A_2，…，A_m，在结点A_i处给定函数值f_i(i=1，…，m)，构造多项式p∈K[X_1，X_2，…，X_n]，满足Lagrange插值条件：p=f_i，i=1，…，m (1)其中X=(X_1，X_2，…，X_n)，与一元情形相似地，本文证明了定理满足插值条件(1)的多项式存在，并且按"序"最低的多项式是唯一的，上述多项式可利用第二章介绍的Lagrange-Hermite插值算法求出，Lagrange插值另一种描述是：按序从低到高给定单项式ω_1，ω_2，…，ω_m，对任意给定的f_1，f_2，…，f_m，构造多项式p，满足插值条件：p=sum from i=1 to m=Ai=f_i，i=1，…，m (2)如果插值多项式p存在且唯一，则称插值问题适定。

In this paper, multivariate Lagrange interpolation problems are systemically discussed,and through amending Cramer Strange Proposition and proving interpolation theory, the geometry construction of the properly posed sets of nodes along the plane algebraic curve is studied, at the same time, a number of practical constructing methods are given.

本文系统阐述了多元Lagrange插值问题，通过修正Cramer奇论并用插值法加以证明，进一步对沿平面代数曲线插值唯一可解点组的几何结构进行深入的研究，同时得到了一些实用性较强的构造方法。

As an application of the interpolation criteria, the interpolation formula of bandlimited transmission signals which satisfied the condition of the Nyquist first criteria has been presented.

在信号插值问题上，首先利用基本的信号模型和Nyquist第一准则，导出了一个插值准则，作为插值准则的应用，对满足Nyquist第一准则的带限传输信号给出了一个通用的插值公式，以升余弦滚降函数为成形滤波函数的限带传输信号作为一个特例，通过计算机仿真验证了插值算法的可行性和有效性。

The problem of Lagrange interpolation of polynomial space in space Rs is studied,and the construction of Lagrange interpolation polynomial in space R1 and space R2 is proposed.

研究空间Rs 中多项式空间中的Lagrange插值问题。给出了R1和R2上Lagrange插值多项式的构造，同时，给出了R2上插值问题的几个例子。另外，给出了矩形网点上的Lagrange插值多项式和三角形网点上的Lagrange插值多项式。讨论了Rs空间中的Lagrange插值多项式及其余项

The multivariate polynomial interpolation problem is a classical mathematical problem that is widely used in many fields, such as multivariate function list, surface design and finite element method. In recent years, multivariate polynomial interpolation has been focused by many people, of which the geometric topological struction of sets of interpolation nodes is also much concerned by us.

多元多项式插值问题是一个十分具有研究意义和实际应用价值的数学问题，它广泛应用于多元函数列表，以及曲面的外形设计和有限元法等诸多领域，近年来多元多项式插值越来越受到人们的广泛关注，其中有关插值结点组的几何拓扑结构问题也是人们十分关注的内容。1998年，梁学章和吕春梅在文献[2]中借助代数几何中的有关理论，进一步讨论了沿无重复分量平面代数曲线上的Lagrange插值问题，并应用Cayley ？

For the interpolation problems in which the nodes are on one or more algebraic manifolds, we pointed out the interpolation bases of the lowest degree w.r.t.a graded term order and gave a new algorithm. We also gave a estimation for the degree of the interpolation polynomial.

我们针对插值节点位于一个或几个代数流形上的插值问题，从理论上指出了插值问题在某一分次序下最低次的插值基并给出了新的算法，同时也对插值多项式的次数作了估计。

In this paper,we discussed the existence,uniqueness,and approximation order of interpolation by bivariate quartic splines with B-Net in three direction meshes .

文[1]、[2]曾讨论了用S41（△mn（1））、S41（△mn（2））作为插值空间的插值问题，[3]讨论了三向剖分域上S31空间的B样条对偶基与拟插值。

As an example to cubic trigonometric polynomial Bézier,the characters of trigonometric polynomial Bézier curve is analyzed,and deduced that cubic trigonometric polynomial Bézier curve is more smooth than cubic polynomial Bézier curve.

Bézier曲线是计算机辅助几何设计中的一类重要曲线，文献[1]介绍了三次Bézier曲线插值，文献[2]介绍了三次Bézier曲线的保凸插值，但难以解决一端曲率为0，另一端曲率比较大的插值问题。

As a application of this theory, we discussed the interpolation problems in which the nodes are ununiform rectanglar grids in real space. We gave the lowest interpolation bases of nodes in rectanglar and sidestep-shaped area. Another application is for finite elements theory. We mainly discussed finite elements of Hermite-Birkhoff type interpolation and gave a new algorithms for a interpolation bases.

作为上述理论的具体应用，我们一方面讨论了插值节点为实空间上的非均匀矩形格点的插值问题，主要给出了矩形区域和阶梯形区域中格点上插值问题的按序最低的插值基的一般公式，另一方面我们研究了多元插值在有限元中的应用，主要是针对Hermite-Birkhoff型插值的有限元给出了计算插值基函数的算法。

First, we introduce and discuss the various methods of multivariate polynomial interpolation in the literature. Based on this study, we state multivariate Lagrange interpolation over again from algebraic geometry viewpoint:Given different interpolation nodes A1,A2 .....,An in the affine n-dimensional space Kn, and accordingly function values fi(i = 1,..., m), the question is how to find a polynomial p K[x1, x2,...,xn] satisfying the interpolation conditions:where X=(x1,X2,....,xn). Similarly with univariate problem, we have provedTheorem If the monomial ordering is given, a minimal ordering polynomial satisfying conditions (1) is uniquely exsisted.Such a polynomial can be computed by the Lagrange-Hermite interpolation algorithm introduced in chapter 2. Another statement for Lagrange interpolation problem is:Given monomials 1 ,2 ,.....,m from low degree to high one with respect to the ordering, some arbitrary values fi(i= 1,..., m), find a polynomial p, such thatIf there uniquely exists such an interpolation polynomial p{X, the interpolation problem is called properly posed.

文中首先对现有的多元多项式插值方法作了一个介绍和评述，在此基础上我们从代数几何观点重新讨论了多元Lagrange插值问题：给定n维仿射空间K~n中两两互异的点A_1，A_2，…，A_m，在结点A_i处给定函数值f_i(i=1，…，m)，构造多项式p∈K[X_1，X_2，…，X_n]，满足Lagrange插值条件：p=f_i，i=1，…，m (1)其中X=(X_1，X_2，…，X_n)，与一元情形相似地，本文证明了定理满足插值条件(1)的多项式存在，并且按&序&最低的多项式是唯一的，上述多项式可利用第二章介绍的Lagrange-Hermite插值算法求出，Lagrange插值另一种描述是：按序从低到高给定单项式ω_1，ω_2，…，ω_m，对任意给定的f_1，f_2，…，f_m，构造多项式p，满足插值条件：p=sum from i=1 to m=Ai=f_i，i=1，…，m (2)如果插值多项式p存在且唯一，则称插值问题适定。

It has been put forward that there exists single Ball point and double Ball points on the symmetrical connecting-rod curves of equilateral mechanisms.

从鲍尔点的形成原理出发，分析对称连杆曲线上鲍尔点的产生条件，提出等边机构的对称连杆曲线上有单鲍尔点和双鲍尔点。

The factory affiliated to the Group primarily manufactures multiple-purpose pincers, baking kits, knives, scissors, kitchenware, gardening tools and beauty care kits as well as other hardware tools, the annual production value of which reaches US$ 30 million dollars.

集团所属工厂主要生产多用钳、烤具、刀具、剪刀、厨具、花园工具、美容套等五金产品，年生产总值3000万美元，产品价廉物美、选料上乘、质量保证，深受国内外客户的青睐