本文選題：理想插值 + 理想投影算子　； 參考：《吉林大學(xué)》2016年博士論文

【摘要】：多項(xiàng)式插值是函數(shù)逼近中常用的方法,也是一個(gè)古老而經(jīng)典的研究問題.隨著科學(xué)技術(shù)的不斷發(fā)展,多項(xiàng)式插值理論現(xiàn)已被廣泛地應(yīng)用在圖像處理、電子通信、控制論、機(jī)械工程等多個(gè)領(lǐng)域.本文感興趣的一類多項(xiàng)式插值是所謂的理想插值,其插值條件只包含有限個(gè)插值節(jié)點(diǎn),每個(gè)節(jié)點(diǎn)上的插值條件泛函由若干賦值泛函與微分算子復(fù)合而成,并且誘導(dǎo)這些微分算子的多項(xiàng)式所構(gòu)成的線性空間是有限維的微分閉子空間.下文稱由插值條件張成的線性泛函空間為插值條件泛函空間.設(shè)F表示特征為零的數(shù)域,F[x]:=F[x1,...,xd]為F上的d元多項(xiàng)式環(huán).F[x]上的投影算子P稱為理想投影算子,如果其核空間為一理想.每個(gè)理想插值問題都可以由一個(gè)理想投影算子P描述：P的對(duì)偶的像空間恰為插值問題的插值條件泛函空間,P的像空間即為插值空間.Lagrange插值是一類最簡單的理想插值問題,其對(duì)應(yīng)的理想投影算子稱為Lagrange投影算子.在一元情況下,所有的理想投影算子均為Lagrange投影算子的逐點(diǎn)極限.這個(gè)結(jié)論在某些多元情況下也成立.因此de Boor定義Hermite投影算子為Lagrange投影算子的極限de Boor曾猜想所有的多元復(fù)理想投影算子均為Hermite投影算子.然而隨后Shekhtman針對(duì)三元以上情形給出其猜想的反例,所以判斷一個(gè)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投影算子列就成為人們關(guān)心的問題.本文將針對(duì)一個(gè)給定的理想插值問題對(duì)應(yīng)的理想投影算子,考慮如何計(jì)算逼近它的Lagrange投影算子列(如果存在),稱這個(gè)問題為理想插值算子的離散逼近問題.為簡便計(jì),也稱為(理想插值的)離散逼近問題或離散問題.本文利用代數(shù)幾何工具并結(jié)合微分閉子空間的結(jié)構(gòu)分析,研究了理想插值算子離散逼近中的若干問題.主要工作如下.1.對(duì)一般的理想投影算子給出了一個(gè)離散逼近算法.理想插值的離散等價(jià)于插值條件的離散,而插值條件由所謂的“微分閉子空間”描述.因此理想插值算子的離散可轉(zhuǎn)化為每個(gè)節(jié)點(diǎn)上微分閉子空間誘導(dǎo)的微分算子的離散,后者簡稱為微分閉子空間的離散逼近問題.因?yàn)閷?duì)一個(gè)理想插值問題,如果每個(gè)點(diǎn)上插值條件泛函空間中的微分算子都可以離散,那么整個(gè)理想插值算子就可以離散,所以以后我們將只考慮一個(gè)點(diǎn)上的離散問題.具體地,當(dāng)給定節(jié)點(diǎn)z及其相應(yīng)的s+1維微分閉子空間Qz(?)F[x]時(shí),研究如何計(jì)算s+1個(gè)點(diǎn)z0(h),…,zs(h),使得其中δz表示z點(diǎn)處的賦值泛函,q(D):=g(D1,...,Dd)表示由q誘導(dǎo)的微分算子,Dj:=(?)/(?)xj表示關(guān)于xj的微分算子,j=1,...,d.稱z0(h),...,zs(h)為離散節(jié)點(diǎn).本文對(duì)插值條件中每個(gè)節(jié)點(diǎn)相應(yīng)的微分閉子空間分別考慮,將離散問題轉(zhuǎn)化為非線性方程組的求解問題.如果最后得到的方程組有解,則輸出相應(yīng)的離散節(jié)點(diǎn).進(jìn)而證明了對(duì)于給定的理想投影算子,如果每個(gè)點(diǎn)上的插值條件都可以離散,則給定的理想投影算子為Hermite投影算子.2.研究了二階微分閉子空間Q2的離散逼近問題.對(duì)于任意一個(gè)多項(xiàng)式線性空間,將基底中的多項(xiàng)式按某個(gè)單項(xiàng)序?qū)懗删仃嚨男问?并對(duì)其進(jìn)行Gauss-Jordan消去,得到的新矩陣就對(duì)應(yīng)原線性空間的另一組基底,稱其為約化基.以后總假定多項(xiàng)式線性子空間的基都是約化基.本文首先研究了特殊的二階微分閉子空間Q2:=span{1,p1(1),...,pm1(1),p(2))的結(jié)構(gòu),其中上角標(biāo)表示多項(xiàng)式的次數(shù).利用變量替換,可以得到Q2約化基中所有一次多項(xiàng)式的一般形式,進(jìn)而可以得到p(2)的結(jié)構(gòu).再利用類似的討論得到一般的二階微分閉子空間Q2的結(jié)構(gòu).然后給出了空間Q2基底中一次多項(xiàng)式對(duì)應(yīng)的離散點(diǎn)集.最后利用已有的一階離散節(jié)點(diǎn),給出了空間δzQ2(D)可以被離散的一個(gè)充分條件.3.解決了寬度為1的微分閉子空間的離散逼近問題.本文首先討論了寬度為1的微分閉子空間結(jié)構(gòu)的另一種等價(jià)表示.然后利用這種等價(jià)表示,給出了此類微分閉子空間對(duì)應(yīng)的兩組離散節(jié)點(diǎn),從而證明了其對(duì)應(yīng)的理想投影算子為Hermite投影算子.4.研究了復(fù)數(shù)域上一般的二元理想插值的離散逼近問題Shekhtman利用代數(shù)幾何工具證明了二元理想投影算子均為Hermite投影算子.本文基于Shekhtman的理論,在假定給定插值節(jié)點(diǎn)上一般插值條件的前提下,給出了解決二元離散逼近問題的構(gòu)造性算法.文中首先針對(duì)單點(diǎn)的理想插值問題,給出一個(gè)計(jì)算由插值條件確定的理想的約化Grobner基算法,進(jìn)而可以求得相應(yīng)的乘法矩陣.然后利用Jordan標(biāo)準(zhǔn)型和一元有理插值方法來計(jì)算離散逼近問題的離散節(jié)點(diǎn).最后就二元寬度為1的微分閉子空間的離散逼近問題給出其對(duì)應(yīng)的一組離散節(jié)點(diǎn).5.利用笛卡爾張量分析了一般的n階微分閉子空間Qn的結(jié)構(gòu).這里Qn(?){f∈F[x]:deg(f)≤n)并且Qn中至少含有一個(gè)n次多項(xiàng)式.設(shè)Qn表示Qn中次數(shù)小于n的多項(xiàng)式集合.與二階情況類似,當(dāng)給定空間Qn時(shí),Qn中的n次多項(xiàng)式具有相同的結(jié)構(gòu),所以不失一般性,可以假設(shè)Qn中只含一個(gè)n次多項(xiàng)式.本文首先研究了Q3=spa{1,p1(1),…,pm1(1),p1(2),…,pm2(2),p(3))中p(3)的結(jié)構(gòu),這里Q3基底中的多項(xiàng)式均為齊次多項(xiàng)式.因?yàn)镽d上的n階對(duì)稱張量構(gòu)成的空間同構(gòu)于全體d元n次齊次多項(xiàng)式構(gòu)成的空間,所以可以用對(duì)稱張量來表示齊次多項(xiàng)式.即任意的三次齊次多項(xiàng)式p(3)都對(duì)應(yīng)一個(gè)三階對(duì)稱笛卡爾張量B(3)∈Rd(?)Rd(?)Rd本文首先證明了B(3)可以寫成由所謂的“關(guān)聯(lián)矩陣”構(gòu)成的張量與Q3中一次多項(xiàng)式構(gòu)成的矩陣的內(nèi)積,然后給出了B(3)中元素的自由度.類似地我們討論了更高階微分閉子空間Qn,n3,中的n次多項(xiàng)式與其中一次多項(xiàng)式的聯(lián)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.

【學(xué)位授予單位】：吉林大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2016
【分類號(hào)】：O174.41

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