理想插值算子离散逼近中若干问题的研究

发布时间：2018-04-23 21:03

本文选题：理想插值 + 理想投影算子　；参考：《吉林大学》2016年博士论文

【摘要】：多项式插值是函数逼近中常用的方法,也是一个古老而经典的研究问题.随着科学技术的不断发展,多项式插值理论现已被广泛地应用在图像处理、电子通信、控制论、机械工程等多个领域.本文感兴趣的一类多项式插值是所谓的理想插值,其插值条件只包含有限个插值节点,每个节点上的插值条件泛函由若干赋值泛函与微分算子复合而成,并且诱导这些微分算子的多项式所构成的线性空间是有限维的微分闭子空间.下文称由插值条件张成的线性泛函空间为插值条件泛函空间.设F表示特征为零的数域,F[x]:=F[x1,...,xd]为F上的d元多项式环.F[x]上的投影算子P称为理想投影算子,如果其核空间为一理想.每个理想插值问题都可以由一个理想投影算子P描述：P的对偶的像空间恰为插值问题的插值条件泛函空间,P的像空间即为插值空间.Lagrange插值是一类最简单的理想插值问题,其对应的理想投影算子称为Lagrange投影算子.在一元情况下,所有的理想投影算子均为Lagrange投影算子的逐点极限.这个结论在某些多元情况下也成立.因此de Boor定义Hermite投影算子为Lagrange投影算子的极限de Boor曾猜想所有的多元复理想投影算子均为Hermite投影算子.然而随后Shekhtman针对三元以上情形给出其猜想的反例,所以判断一个多元理想投影算子是否是Hermite投影算子;如果它是Hermite投影算子,如何得到逼近它的Lagrange投影算子列就成为人们关心的问题.本文将针对一个给定的理想插值问题对应的理想投影算子,考虑如何计算逼近它的Lagrange投影算子列(如果存在),称这个问题为理想插值算子的离散逼近问题.为简便计,也称为(理想插值的)离散逼近问题或离散问题.本文利用代数几何工具并结合微分闭子空间的结构分析,研究了理想插值算子离散逼近中的若干问题.主要工作如下.1.对一般的理想投影算子给出了一个离散逼近算法.理想插值的离散等价于插值条件的离散,而插值条件由所谓的“微分闭子空间”描述.因此理想插值算子的离散可转化为每个节点上微分闭子空间诱导的微分算子的离散,后者简称为微分闭子空间的离散逼近问题.因为对一个理想插值问题,如果每个点上插值条件泛函空间中的微分算子都可以离散,那么整个理想插值算子就可以离散,所以以后我们将只考虑一个点上的离散问题.具体地,当给定节点z及其相应的s+1维微分闭子空间Qz(?)F[x]时,研究如何计算s+1个点z0(h),…,zs(h),使得其中δz表示z点处的赋值泛函,q(D):=g(D1,...,Dd)表示由q诱导的微分算子,Dj:=(?)/(?)xj表示关于xj的微分算子,j=1,...,d.称z0(h),...,zs(h)为离散节点.本文对插值条件中每个节点相应的微分闭子空间分别考虑,将离散问题转化为非线性方程组的求解问题.如果最后得到的方程组有解,则输出相应的离散节点.进而证明了对于给定的理想投影算子,如果每个点上的插值条件都可以离散,则给定的理想投影算子为Hermite投影算子.2.研究了二阶微分闭子空间Q2的离散逼近问题.对于任意一个多项式线性空间,将基底中的多项式按某个单项序写成矩阵的形式,并对其进行Gauss-Jordan消去,得到的新矩阵就对应原线性空间的另一组基底,称其为约化基.以后总假定多项式线性子空间的基都是约化基.本文首先研究了特殊的二阶微分闭子空间Q2:=span{1,p1(1),...,pm1(1),p(2))的结构,其中上角标表示多项式的次数.利用变量替换,可以得到Q2约化基中所有一次多项式的一般形式,进而可以得到p(2)的结构.再利用类似的讨论得到一般的二阶微分闭子空间Q2的结构.然后给出了空间Q2基底中一次多项式对应的离散点集.最后利用已有的一阶离散节点,给出了空间δzQ2(D)可以被离散的一个充分条件.3.解决了宽度为1的微分闭子空间的离散逼近问题.本文首先讨论了宽度为1的微分闭子空间结构的另一种等价表示.然后利用这种等价表示,给出了此类微分闭子空间对应的两组离散节点,从而证明了其对应的理想投影算子为Hermite投影算子.4.研究了复数域上一般的二元理想插值的离散逼近问题Shekhtman利用代数几何工具证明了二元理想投影算子均为Hermite投影算子.本文基于Shekhtman的理论,在假定给定插值节点上一般插值条件的前提下,给出了解决二元离散逼近问题的构造性算法.文中首先针对单点的理想插值问题,给出一个计算由插值条件确定的理想的约化Grobner基算法,进而可以求得相应的乘法矩阵.然后利用Jordan标准型和一元有理插值方法来计算离散逼近问题的离散节点.最后就二元宽度为1的微分闭子空间的离散逼近问题给出其对应的一组离散节点.5.利用笛卡尔张量分析了一般的n阶微分闭子空间Qn的结构.这里Qn(?){f∈F[x]:deg(f)≤n)并且Qn中至少含有一个n次多项式.设Qn表示Qn中次数小于n的多项式集合.与二阶情况类似,当给定空间Qn时,Qn中的n次多项式具有相同的结构,所以不失一般性,可以假设Qn中只含一个n次多项式.本文首先研究了Q3=spa{1,p1(1),…,pm1(1),p1(2),…,pm2(2),p(3))中p(3)的结构,这里Q3基底中的多项式均为齐次多项式.因为Rd上的n阶对称张量构成的空间同构于全体d元n次齐次多项式构成的空间,所以可以用对称张量来表示齐次多项式.即任意的三次齐次多项式p(3)都对应一个三阶对称笛卡尔张量B(3)∈Rd(?)Rd(?)Rd本文首先证明了B(3)可以写成由所谓的“关联矩阵”构成的张量与Q3中一次多项式构成的矩阵的内积,然后给出了B(3)中元素的自由度.类似地我们讨论了更高阶微分闭子空间Qn,n3,中的n次多项式与其中一次多项式的联系.
[Abstract]:Polynomial interpolation is a common method in function approximation. It is also an ancient and classical research problem. With the continuous development of science and technology, polynomial interpolation theory has been widely used in many fields, such as image processing, electronic communication, cybernetics, mechanical engineering and so on. A kind of polynomial interpolation that is interested in this paper is the so-called ideal interpolation. The interpolation condition consists of only finite interpolation nodes. The interpolation condition functional on each node is composed of several assignment functionals and differential operators, and the linear space formed by the polynomial of these differential operators is a finite dimensional differential closed subspace. Let F represent a number field with zero characteristic, F[x]: =F[x1,..., xd] is a projection operator on the D element polynomial ring.F[x] on F, which is called an ideal projection operator. If its kernel space is an ideal, every ideal interpolation problem can be described by an ideal projection operator P: the dual image space of P is exactly the interpolation condition of the interpolation problem. Functional space, the image space of P is the interpolation space.Lagrange interpolation is the simplest kind of ideal interpolation problem, and its corresponding ideal projection operator is called Lagrange projection operator. In the case of one element, all the ideal projection operators are the point by point limit of the Lagrange projection operator. This conclusion is also established in some multivariate cases. So de Boor defines the limit de Boor of the projection operator of the Hermite projection operator as the Lagrange projection operator. It has been conjectured that all the multivariate complex ideal projection operators are Hermite projection operators. However, Shekhtman then gives the counterexample of its conjecture on the case of more than three yuan, and determines whether a multivariate ideal projection operator is a Hermite projection operator; if it is Hermite The projection operator, how to get the approximation of its Lagrange projection operator is a concern. This paper will consider an ideal projection operator for a given ideal interpolation problem and consider how to calculate the approximation of its Lagrange projection operator (if existence), which is called the discrete approximation problem of the ideal interpolation operator. It is also known as the discrete approximation problem or discrete problem of (ideal interpolation). In this paper, some problems in the discrete approximation of ideal interpolating operators are studied by using algebraic geometric tools and combining the structural analysis of differential closed subspaces. The main work is as follows:.1. gives a discrete approximation algorithm for the general ideal projector. Ideal interpolation The discrete is equivalent to the discrete interpolation condition, and the interpolation condition is described by the so-called "differential closed subspace". Therefore, the discrete of the ideal interpolation operator can be transformed into the discrete differential operator induced by the differential closed subspace on each node, and the latter is referred to as the discrete approximation problem of the differential closed subspace. The differential operators in the interpolation conditional functional space at each point can be discrete, then the whole ideal interpolation operator can be discrete, so we will consider the discrete problem on one point in the future. Specifically, when the given node Z and its corresponding s+1 dimensional differential closed subspace Qz (?) F[x], we study how to calculate the s+1 point Z0 (H),... ZS (H), which makes delta Z represent the assignment functional at z point, q (D): =g (D1,..., Dd) is a differential operator induced by Q, Dj:= (?) / (?) XJ represents the differential operator, which is a discrete node. This paper considers the differential closed subspace corresponding to each node in the interpolation condition, and transforms the discrete problem into nonlinear If the final equation group has solutions, the corresponding discrete node is output. And it is proved that for a given ideal projection operator, if the interpolation conditions on each point can be discrete, then the given ideal projection operator is Hermite projection operator.2. to study the discrete approximation of the two order differential closed subspace Q2. For any polynomial linear space, the polynomials in the base are written in the form of a single order in the form of a single order, and they are eliminated by Gauss-Jordan. The new matrix is corresponding to the other base of the original linear space, which is called the reductive basis. A special structure of two order differential closed subspaces Q2:=span{1, P1 (1),..., PM1 (1), P (2)), in which the upper corner marks the number of polynomials. By substitution of variables, the general form of all the polynomial in the Q2 reduct can be obtained, and then the structure of P (2) can be obtained. Then the general two order differential closed subspace is obtained by the similar discussion. The structure of Q2. Then the discrete point set corresponding to a polynomial in the space Q2 base is given. Finally, using the existing first order discrete nodes, the discrete approximation problem of the differential closed subspace with the width of 1 can be solved by a sufficient condition.3. that can be discrete. This paper first discusses the differential closed subspace with a width of 1. Another equivalent representation of the structure, and then using this equivalent representation, two groups of discrete nodes corresponding to this kind of differential closed subspace are given, and it is proved that the corresponding ideal projection operator is Hermite projection operator.4. to study the discrete approximation problem of the general two element ideal interpolation in the complex field. Shekhtman uses Algebraic Geometric tools. It is clear that all the two element ideal projection operators are Hermite projection operators. Based on the theory of Shekhtman, this paper gives a constructive algorithm for solving the two element discrete approximation problem on the premise of assuming the general interpolation conditions on a given interpolating node. The corresponding multiplication matrix can be obtained by the reductive Grobner based algorithm, and then the discrete nodes of the discrete approximation problem are calculated using the Jordan standard type and the univariate rational interpolation method. Finally, the discrete approximation problem of the differential closed subspace with the two element width of 1 is given by the Cartesian tensor analysis of its corresponding discrete node.5.. The structure of the general n order differential closed subspace Qn. Here Qn (?) {f F[x]: DEG (f) < n) and Qn contains at least one n subpolynomial. There is a n degree polynomial. In this paper, we first study Q3=spa{1, P1 (1),... PM1 (1), P1 (2),... The structure of P (3) in PM2 (2) and P (3)), the polynomials in the Q3 base are homogeneous polynomials. Because the space of the n order symmetric tensor on Rd is isomorphic to the space made up of all d elements n order polynomial, so the symmetric tensor can be used to express the homogeneous polynomial. That is, any three homogeneous polynomial P (3) corresponds to a three order symmetrical flute. Carle tensor B (3) Rd (?) Rd (?) Rd this paper first proves that B (3) can be written as the inner product of the tensor made up of so-called "correlation matrix" and the first polynomial in the Q3, and then gives the degree of freedom of the element in B (3). Similarly, we discuss the higher order differential closed subspace Qn, N3, the N sub polynomial in the N3, and one of the multiple polynomials. The type of connection.

【学位授予单位】：吉林大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O174.41

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