本文關(guān)鍵詞： 理想插值 理想投影算子 Gr(?)bner基 誤差公式 離散化　出處：《吉林大學(xué)》2016年博士論文　論文類型：學(xué)位論文

【摘要】：理想插值最早由數(shù)學(xué)家Birkhoff提出,用來研究一般的多元多項(xiàng)式插值問題.理想插值是一種被插函數(shù)為多項(xiàng)式的線性插值格式,其可看成是經(jīng)典的一元Lagrange插值與Hermite插值在多元情形下的推廣.具體來說,理想插值由理想投影算子確定.理想投影算子是多項(xiàng)式空間到自身的線性冪等算子,其核恰為一多項(xiàng)式理想.在理想插值中,理想投影算子的像空間為插值空間,理想投影算子的對(duì)偶的像空間為插值條件泛函空間.插值條件泛函空間由一組插值節(jié)點(diǎn),以及每個(gè)節(jié)點(diǎn)上相應(yīng)的賦值泛函與由有限維微分閉子空間所定義的微分算子的復(fù)合構(gòu)成.微分閉子空間是由多項(xiàng)式構(gòu)成的線性空間,并且其對(duì)求導(dǎo)運(yùn)算是封閉的.由于“微分閉”的概念是對(duì)一元Hermite插值條件中“連續(xù)階”導(dǎo)數(shù)要求的推廣,所以理想插值包含了經(jīng)典的Lagrange插值與Hermite插值,其中Lagrange插值對(duì)應(yīng)的理想投影算子稱為L(zhǎng)agrange投影算子.2005年,de Boor在他的理想插值綜述中提到下列問題,其一：理想投影算子是否具有統(tǒng)一的誤差結(jié)構(gòu)表達(dá)式；其二：哪些理想投影算子具有“好”誤差公式；其三：若一理想投影算子為Hermite投影算子,如何計(jì)算逼近它的Lagrange投影算子列.到目前為止,這些問題仍然是理想插值中的研究熱點(diǎn).為簡(jiǎn)便起見,我們稱前兩個(gè)問題為理想插值的誤差公式問題,稱最后一個(gè)問題為理想插值的離散化問題.本文將利用代數(shù)幾何的理論知識(shí)研究上述問題,并給出一些理論結(jié)果.主要工作如下：1.給出了理想投影算子統(tǒng)一的誤差結(jié)構(gòu)表達(dá)式.一元理想投影算子的誤差的結(jié)構(gòu)形式簡(jiǎn)單優(yōu)美.為將其推廣到多元情形,de Boor提出了理想投影算子的“好”誤差公式的概念.“好”誤差公式是一種誤差結(jié)構(gòu)表達(dá)式,具體說,是指存在齊次多項(xiàng)式Hj和線性算子q使得插值誤差可以表示為f-Pf＝ ∑j=1mCj(Hj(D)f)hj且滿足正交條件Hj(D)hk=δj,k,其中f為被插多項(xiàng)式函數(shù),P為理想投影算子,馬(D)為微分算子,{h1,...,hm}為理想kerP的理想基de Boor曾猜測(cè)所有理想投影算子都具有“好”誤差公式,但隨后Shekhtman給出了一個(gè)二元情形下的反例,并斷言大多數(shù)理想投影算子都不具有“好”誤差公式.我們研究了理想投影算子的誤差公式的代數(shù)結(jié)構(gòu),在“好”誤差公式的基礎(chǔ)上,提出了“一般”型誤差公式的概念.然后利用理想的約化理論,證明了所有理想投影算子的核空間的字典序下的約化Grobner基都支撐“一般”型誤差公式.最后利用B樣條理論,給出了Shekhtman反例的“一般”型誤差公式的具體表達(dá)式.2.給出了一類Lagrange投影算子的“好”誤差公式的具體表達(dá)式.前面我們提到不是所有理想投影算子都具有“好”誤差公式.到現(xiàn)在為止,人們對(duì)“好”誤差公式的存在性的研究取得了一定進(jìn)展Shekhtman證明了特殊幾何分布節(jié)點(diǎn)上的理想投影算子具有“好”誤差公式,李U喼っ髁司叻篏robner基的理想投影算子有“好”誤差公式,de Boor給出了具張量積節(jié)點(diǎn)和滿足GC條件節(jié)點(diǎn)的Lagrange投影算子的“好”誤差公式的具體表達(dá)式.受這些工作的啟發(fā),我們研究了一類特殊理想投影算子的誤差公式.針對(duì)Cartesian點(diǎn)集上的Lagrange投影算子,首先利用差商算法,給出插值余項(xiàng).然后將插值余項(xiàng)整理成差商形式的“好”誤差公式.最后利用差商與樣條積分的關(guān)系,給出了“好”誤差公式的具體表達(dá)式.3.研究了一類二元Hermite投影算子的離散化問題.當(dāng)人們推廣一個(gè)概念時(shí),一般會(huì)保留原有的結(jié)構(gòu)屬性.在一元情形下,Hermite插值是Lagrange插值的極限形式.這一事實(shí)啟發(fā)de Boor定義Hermite投影算子為L(zhǎng)agrange投影算子的極限.雖然一元理想投影算子都是Hermite投影算子,并且這個(gè)結(jié)論在某些多元情形下也成立,但已有例子表明存在非Hermite的多元理想投影算子.所以判斷一個(gè)理想投影算子是否為Hermite投影算子,以及如何計(jì)算逼近Hermite投影算子的Lagrange投影算子列是人們十分關(guān)心的問題.圍繞這個(gè)問題(理想插值的離散化問題),de Boor和Shekhtman證明了二元理想投影算子都是Hermite投影算子,并給出了一種計(jì)算其相應(yīng)Lagrange投影算子列的方法.但是方法本身復(fù)雜度高,不易于實(shí)現(xiàn).我們研究了一類特殊的二元Hermite投影算子,其插值條件泛函為δξοΡ(n)(D),Ρ(n):=Fn[x,y](?)spanF{pn},其中δζ為ζ點(diǎn)處的賦值泛函,D為微分算符,Fn[x,y]為次數(shù)小于n的二元多項(xiàng)式集合,pn為一任意的二元n次多項(xiàng)式.針對(duì)此類Hermite投影算子,我們給出了一種計(jì)算Lagrange投影算子列的方法,該方法簡(jiǎn)單有效并且?guī)缀醪挥萌魏斡?jì)算代價(jià).
[Abstract]:The ideal interpolation was first proposed by mathematician Birkhoff, multivariate polynomial interpolation is used to study the problem in general. The ideal interpolation is an interpolating function for linear interpolation polynomial, which can be regarded as a generalization of the classical Lagrange interpolation and Hermite interpolation in the multivariate case. Specifically, the ideal interpolation is determined by the ideal projection operator. The ideal projection operator is idempotent operator's linear polynomial space, the kernel is a polynomial ideal. The ideal interpolation, the ideal projection operator like space interpolation, dual ideal projection operator like space interpolation conditions. The functional space interpolation functional space consists of a set of interpolation nodes, and the composition of each the corresponding node assignment by finite dimensional differential functionals and closed subspaces defined by differential operators. Differential closed subspace is constructed by polynomials of the linear space, And it is closed to the derivative operations. Due to the "differential closed" concept of Hermite interpolating condition "continuous order derivative" requirements of the promotion, so the ideal interpolation contains the classical Lagrange interpolation and Hermite interpolation, which corresponds to the ideal projection Lagrange interpolation operator called Lagrange projection operator.2005, de Boor mentioned the following questions in his review: the ideal interpolation error expression is the ideal structure of projection operator is unified; second: what is the "good" ideal projection operator error formula; third: if an ideal projection operator for Hermite projection operator, how to calculate the approximation of Lagrange projection operator of it. So far, the problem is still a hot topic in the ideal interpolation. For simplicity, we call the first two questions for the problem of the ideal interpolation error formula, called the last question For the discretization of the ideal interpolation. The knowledge of algebraic geometry theory this paper will use, some theoretical results are also given. The main work is as follows: 1. gives the error structure expression of ideal projection operator. A unified structure error ideal projection operator is simple is beautiful. Which could be generalized to the multivariate case, de Boor put forward the concept of the ideal projection operator of "good" error formula. "Good" is a kind of error formula of error structure expression, specifically, refers to the existence of homogeneous polynomial Hj and linear operator Q so that the interpolation error can be expressed as f-Pf = j=1mCj (Hj (D) F) HJ and orthogonal Hj (D) hk= J, K F, which is inserted polynomial function, P is the ideal projection operator, MA (D) for the differential operator, {h1,..., hm} is the ideal Boor ideal kerP based de had to guess all the ideal projection operator has "good" error formula But then, Shekhtman gave a counterexample of two yuan case, and asserts that the most ideal projection operator is not "good" error formula. We study the algebraic structure of the error formula of ideal projection operator, based on "good" error formula, puts forward the concept of "a" type of error formula and then use the ideal reduction theory, proved that the reduced Grobner based kernel space all ideal projection operator in lexicographic order under the support "general" type error formula. Finally using B spline theory, gives the Shekhtman a counterexample "general" type error formula of the specific expression of.2. presents a Lagrange projection operator "good" error formula of specific expression. We mentioned is not all ideal projection operator has good error formula. Until now, the people of the "good" error formula of the existence of research The progress of Shekhtman proved that the ideal projection operator special geometric distribution node has a "good" error formula, "good" error formula of ideal projection operator Li U after our Qian Le robner condyle of Worcestershire based, de Boor provides a specific expression with tensor product nodes and meet the Lagrange GC node projection operator the "good" error formula. Inspired by their work, we study the error formula of projection operator of a special kind of ideal for Lagrange projection operator Cartesian points on the first use of difference algorithm, is presented. Then the interpolation remainder interpolation remainder sorting into difference quotient "good" error formula. Finally the relationship the difference with the spline integral, the discrete problem specific expression.3. "error formula is studied for a class of two yuan Hermite projection operator is given. When people promote a concept, The general structure will retain the properties of the original. In one yuan case, Hermite interpolation is an extreme form of Lagrange interpolation. This fact inspired de Boor definition of Hermite projection operator to limit Lagrange projection operator. Although the ideal projection operator is Hermite projection operator, and this conclusion in some multivariate case was also established, but there are examples of multiple ideal projection operator in non Hermite. So the judge an ideal projection operator is Hermite projection operator, and how to calculate the Lagrange projection approximation of Hermite projection operator column is the problem that people cares very. Around this problem (discrete ideal interpolation), De, Boor and Shekhtman proved that two the ideal element projection operator is Hermite projection operator, a method of calculating the corresponding Lagrange projection operator sequence is given. But the method itself high complexity, Not easy to implement. We study a special class of two yuan Hermite projection operator, the interpolation condition for the delta zeta functional / P (n) (D), P (n): =Fn[x, y] (?) spanF{pn}, which is at the point of the delta zeta zeta functional assignment, D differential operator, Fn[x. Two yuan is smaller than the number of polynomial y] n collection, PN to two yuan n polynomial. For such an arbitrary Hermite projection operator, we give a method of calculating the Lagrange projection operator, the method is simple and effective and almost without any cost.

【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)】：O241.3

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