多元直交多项式与数值积分公式
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摘要
多元直交多项式和数值积分公式是当前数值逼近研究领域中的热门研究方向，而且在计算几何、科学计算、调和分析、特殊函数以及概率论与统计等诸多数学领域中有着重要应用。它们之间亦有着非常深刻的联系。本文主要针对这两个专题进行研究。本文主要研究：任意区域上的多元直交多项式一般性质及有关数值积分公式的构造问题。主要工作如下：
(1)文献表明连分式是最初研究直交多项式的主要工具之一，但连分式的重要性在较长时间并未引起人们的足够重视。一个很重要的原因是它很难向多元推广。Stieltjes定理揭示了连分式与直交多项式之间的重要关系．罗钟铉教授和王仁宏教授提出了不变因子的概念，从而把一元的Stieltjes定理推广到了二元，并且得到了类似于一元的渐进展开公式，其中不变因子替代了一元直交多项式的作用。本文在的基础上进一步研究了不变因子的性质，特别是它的零点性质。首先证明了不变因子的零点一定是相应的截断Jacobi矩阵的一个特征值，从而是实的；特别地，当不变因子不存在重根时，二者除相差一常数因子外是相等的。另外对于不变因子的一个零点，它都对应着一个直交多项式分解，即存在着一个直交多项式可以被分解为一个一次多项式和一个低一次的多项式的乘积。由此得到了不变因子的类似于一元直交多项式的零点存在范围。
(2)一元直交多项式的零点性质是直交多项式理论的一个重要组成部分。为了研究多元情形的相应性质，把给定次数的所有直交多项式的公共零点视为一元直交多项式零点的推广。在这种观点下，一元的许多性质得以推广到多元，例如公共零点都是实的，且都是单的等。然而，它的存在范围问题一直没有得到解决。在一元情形下，[a，b]上的直交多项式的零点都在(a，b)内部。多元情形要复杂的多：多元直交多项式的公共零点未必在积分区域内部，一个显然的例子是当积分区域为圆环面时，原点作为所有奇次直交多项式的公共零点却不是圆环上的点。通过对不变因子的研究，本文证明了二元直交多项式的公共零点位于积分区域的凸包内部。这一结论可以看作是对多元直交多项式经典理论的一个补充。
(3)数值积分的研究在理论与应用上都有重要的意义。一元的相关理论都已基本完善，而多元数值积分的构造还有很大的困难。在一元情形下，数值积分的求积结点通常取为直交多项式的零点，而这主要得益于人们对一元直交多项式零点性质的了解。多元情形下，对于绝大多数的积分泛函而言，它的所有n次直交多项式的公共零点并不能构成2n-1次积分公式的求积结点，此时选择一些其它具有直交性质的多项式的公共零点作为求积结点是一个可行的办法，但这样做的一个困难是很难判断这些多项式的公共零点个数。由不变因子的性质知道它可以用来分解直交多项式，特别是对于低次的直交多项式，当n=2时，不变因子的任何一个零点都可以把其中的一个2次直交多项式分解为两个一次多项式的乘积。根据Stroud的结论。两个2次直交多项式有4个交点(非无穷远点)，则这4个交点是3次求积公式的积分结点。显然判断具有上面分解性质的两个2次直交多项式的交点个数问题转化为了判断直线交点问题。利用这一性质，本文给出了四点三次积分公式的一个构造方法，并且给出了详细的构造过程。根据M(?)ller的结论，结点数已经达到最小。值得一提的是，利用本文的方法所构造的数值例子作为反例否定了M(?)ller于2004年给出的重要结论：数值积分公式中的结点数最小时相应数值积分公式的所有权系数为正。这一点已经得到M(?)ller本人的邮件承认。
(4)为了研究多元直交多项式的性质，本文给出了高维情形的不变因子的概念，并且由此给出了多元Stieltjes型定理。另外我们还得到了完全与二元直交多项式相平行的结论，其中包括与Jacobi矩阵的关系，不变因子的零点性质，多元直交多项式的分解性质，公共零点的存在范围等。
(5)乘积型公式是构造最简单且应用非常广泛的一类多元数值积分公式。它的构造原理是通过一个变换把原始积分问题分解为一系列的单重积分问题，从而借助一元的求积公式来求积。它的一个最大缺点是在次数和维数升高时结点数增长太快。鉴于此，本文针对球域上的积分问题给出了一个变换，通过这个变换不仅可以把球域上的积分问题转化为单重积分问题，而且最终的结点数也大为减少。本文构造的的求积公式无论代数精度高低总有一些结点是被重复使用的。
(6)多项式插值与数值积分之间有着密切的联系。为了研究插值适定性问题，梁学章等人关于多元多项式的插值问题给出了一种递规构造的方法—添加代数曲线法。本文考虑求积公式的递归构造方法．本文首先给出了一种添加代数曲线的积分构造法，即通过在给定的代数曲线上选择一些结点得到一个代数精度更高的求积公式。选定一些不同的代数曲线，重复添加曲线过程，这样最终得到了递归构造法。其次，为了减少结点数，事先把这些代数曲线视为一个n次直交多项式所对应的代数曲线的一些分支，利用直交多项式的直交性可以假定以这个直交多项式为权函数的积分已经有了一个O点n-1次的积分公式。由于出发点是从n—1次开始，这样势必减少了大量的结点数。本文以圆盘上的积分为例给出了详细的构造过程，并且得到了许多相应的求积公式，有些公式的结点数已经达到最小。最后，本文还给出了不同权系数下直交多项式之间的一个关系式。
(7)借助多项式理想的相关知识，本文研究了一个事先给定结点的数值积分公式的构造方法。把给定的结点限制为一个低次求积公式的结点，由此进一步得到一个更高次代数精精度的求积公式。如此可得到一个嵌入列，即低次求积公式的结点都是高次积分公式的结点。此类研究的意义在于：多元数值积分的误差分析一直是一个难题，人们通常以两个不同精度的求积公式的差表示积分误差。此时，利用嵌入式积分公式序列将有利于分析和减少计算量。另外，文中还给出了一个二元非插值型积分公式的例子。
Orthogonal polynomials and numerical integration in several variables are two hot problems for discussion in the field of numerical approximation, and have many important applications in computational geometry, scientific computing, harmonic analysis, the theory of special functions, probability and statistics and so on. Furthermore both of them are also closely related. In this paper, some problems about these two topics are discussed, which mainly include some extensions of general properties of one dimensional orthogonal polynomials and the construction of cubature formulae with special structures.
(1) Historically, the orthogonal polynomials originated in the theory of continued fractions. But continued fractions have been gradually abandoned as a starting point for the theory of orthogonal polynomials. One of the important reasons is that it is difficult to be extended. To overcome it, professor Luo Zhongxuan and professor Wang Renhong introduced the concepts of invariant factor for orthogonal polynomials of two variables. By invariant factor, they proved the Stieltjes theorem of two variables, in which invariant factors play an important role. In this paper, some further properties of invariant factor are discussed, especially the properties of its zeros. The following results are presented: Every zero of invariant factor must be an eigenvalue of the corresponding truncated Jacobi matrix, and hence real. Especially, if all the zeros of invariant factor are distinct from each other, then the invariant factor equals to the characteristic polynomial of the corresponding truncated Jacobi matrix up to a constant multiple. Moreover, for every zero of invariant factor, there must exist an orthogonal polynomial factored into two polynomials, from which an zero location of invariant factor, similar to the case of one dimensional orthogonal polynomial, can be obtained.
(2) The properties of zeros for one dimensional orthogonal polynomial are the important part of the general properties. Usually to generalize the corresponding properties, common zeros of orthogonal polynomials are used in the case of several variables. Thus part of theory in one variable can be extended to several variables. But the common zeros location remains unknown. In one variable, all the zeros of orthogonal polynomials are located in the interior of integration interval, which is different from the case of two variables. The common zeros of all multivariate orthogonal polynomials with same total degree can lie outside of the region. An obvious example is the orthogonal polynomials on the two dimensional spherical shell. The original point is the common zero of all orthogonal polynomials with odd degree but lie outside of the spherical shell. Base on the properties of invariant factor we prove that all the common zeros lie in the interior of the convex closed hull of the integration region.
(3) Numerical integration is closely related to orthogonal polynomials, and in one variable it usually takes the zeros of orthogonal polynomials as its knots which benefit from the good properties of univariate orthogonal polynomials. However, in higher dimension we know a litter about this. One of the difficulties is one does not know the exact number of common zeros for a set of polynomials. It follows from the property of invariant factor that for every zero of invariant factor there must exist an orthogonal polynomial which can be factored into a product of two polynomials of lower degree. Especially, for n=2, these two polynomials of lower degree is of degree 1. Stroud proved that if two orthogonal polynomials have four intersections, then these points must constitute a set of knots of a cubature formula of degree 3. Obviously, it is easy to compute the number of the common zeros of two orthogonal polynomials with the factoring property. It is based on this idea that we present a method to construct the cubature formula of degree 3 with 4 knots.
(4) To investigate the properties of multivariate orthogonal polynomials, we extend the concept of invariant factor and get the similar results with two variables. The presented results include multivariate Stieltjes type theorem, the relations among invariant factor, multivariate orthogonal polynomials and Jacobi matrix, and common zeros location of multivariate orthogonal polynomials and so on.
(5) Among various methods of constructing cubature formulae, the method of Cartesian product stands out with its simplicity. Briefly, the method for constructing product formulas is the method of separation of variables. If we can find a (nonlinear) transform which transforms the monomial integral into the product of d single integrals and if suitable formulas are known for these single integrals, then one can combine these formulas to give a formula. The most undesired property of product formulas is that the number of points increases very rapidly as d increases. To avoid it partly, we present a proper transformation which transforms the initial integration into the product of single integrations and reduces the number of knots greatly. Furthermore, some knots are used repeatedly by the cubature formulae of any degree.
(6) Polynomial interpolation and quadrature formula are closely related. To investigate the interpolation problem of two variables, Liang presented a kind of recursive method of constructing properly posed set of nodes. This paper consider the corre- sponding recursive method of construction of numericai integration. First we present a construction method of cubature formulae with some knots along the selected algebraic curve. By selecting different algebraic curves and repeating the above process, we will obtain a cubature formula and then a recursive method. To reduce the number of the knots, we take the polynomial corresponding to the selected curve as an orthogonal polynomial. Thus the number of the knots of final cubature formula will be reduced greatly. Taking the unit disk as an example, a detailed construction process is given. Some cubature formula have gotten to the lower bound.
(7) By the knowledge of polynomial ideal, we present a construction method of cubature formula with some fixed points. If we take the fixed points as the nodes of a cubature of lower degree and get a new cubature of higher degree, then an embedded formula sequence is obtained. This kind of embedded sequence can be used to estimate the errors. Furthermore, a noninterpolationary cubature formula is presented.
(1)文献表明连分式是最初研究直交多项式的主要工具之一，但连分式的重要性在较长时间并未引起人们的足够重视。一个很重要的原因是它很难向多元推广。Stieltjes定理揭示了连分式与直交多项式之间的重要关系．罗钟铉教授和王仁宏教授提出了不变因子的概念，从而把一元的Stieltjes定理推广到了二元，并且得到了类似于一元的渐进展开公式，其中不变因子替代了一元直交多项式的作用。本文在的基础上进一步研究了不变因子的性质，特别是它的零点性质。首先证明了不变因子的零点一定是相应的截断Jacobi矩阵的一个特征值，从而是实的；特别地，当不变因子不存在重根时，二者除相差一常数因子外是相等的。另外对于不变因子的一个零点，它都对应着一个直交多项式分解，即存在着一个直交多项式可以被分解为一个一次多项式和一个低一次的多项式的乘积。由此得到了不变因子的类似于一元直交多项式的零点存在范围。
(2)一元直交多项式的零点性质是直交多项式理论的一个重要组成部分。为了研究多元情形的相应性质，把给定次数的所有直交多项式的公共零点视为一元直交多项式零点的推广。在这种观点下，一元的许多性质得以推广到多元，例如公共零点都是实的，且都是单的等。然而，它的存在范围问题一直没有得到解决。在一元情形下，[a，b]上的直交多项式的零点都在(a，b)内部。多元情形要复杂的多：多元直交多项式的公共零点未必在积分区域内部，一个显然的例子是当积分区域为圆环面时，原点作为所有奇次直交多项式的公共零点却不是圆环上的点。通过对不变因子的研究，本文证明了二元直交多项式的公共零点位于积分区域的凸包内部。这一结论可以看作是对多元直交多项式经典理论的一个补充。
(3)数值积分的研究在理论与应用上都有重要的意义。一元的相关理论都已基本完善，而多元数值积分的构造还有很大的困难。在一元情形下，数值积分的求积结点通常取为直交多项式的零点，而这主要得益于人们对一元直交多项式零点性质的了解。多元情形下，对于绝大多数的积分泛函而言，它的所有n次直交多项式的公共零点并不能构成2n-1次积分公式的求积结点，此时选择一些其它具有直交性质的多项式的公共零点作为求积结点是一个可行的办法，但这样做的一个困难是很难判断这些多项式的公共零点个数。由不变因子的性质知道它可以用来分解直交多项式，特别是对于低次的直交多项式，当n=2时，不变因子的任何一个零点都可以把其中的一个2次直交多项式分解为两个一次多项式的乘积。根据Stroud的结论。两个2次直交多项式有4个交点(非无穷远点)，则这4个交点是3次求积公式的积分结点。显然判断具有上面分解性质的两个2次直交多项式的交点个数问题转化为了判断直线交点问题。利用这一性质，本文给出了四点三次积分公式的一个构造方法，并且给出了详细的构造过程。根据M(?)ller的结论，结点数已经达到最小。值得一提的是，利用本文的方法所构造的数值例子作为反例否定了M(?)ller于2004年给出的重要结论：数值积分公式中的结点数最小时相应数值积分公式的所有权系数为正。这一点已经得到M(?)ller本人的邮件承认。
(4)为了研究多元直交多项式的性质，本文给出了高维情形的不变因子的概念，并且由此给出了多元Stieltjes型定理。另外我们还得到了完全与二元直交多项式相平行的结论，其中包括与Jacobi矩阵的关系，不变因子的零点性质，多元直交多项式的分解性质，公共零点的存在范围等。
(5)乘积型公式是构造最简单且应用非常广泛的一类多元数值积分公式。它的构造原理是通过一个变换把原始积分问题分解为一系列的单重积分问题，从而借助一元的求积公式来求积。它的一个最大缺点是在次数和维数升高时结点数增长太快。鉴于此，本文针对球域上的积分问题给出了一个变换，通过这个变换不仅可以把球域上的积分问题转化为单重积分问题，而且最终的结点数也大为减少。本文构造的的求积公式无论代数精度高低总有一些结点是被重复使用的。
(6)多项式插值与数值积分之间有着密切的联系。为了研究插值适定性问题，梁学章等人关于多元多项式的插值问题给出了一种递规构造的方法—添加代数曲线法。本文考虑求积公式的递归构造方法．本文首先给出了一种添加代数曲线的积分构造法，即通过在给定的代数曲线上选择一些结点得到一个代数精度更高的求积公式。选定一些不同的代数曲线，重复添加曲线过程，这样最终得到了递归构造法。其次，为了减少结点数，事先把这些代数曲线视为一个n次直交多项式所对应的代数曲线的一些分支，利用直交多项式的直交性可以假定以这个直交多项式为权函数的积分已经有了一个O点n-1次的积分公式。由于出发点是从n—1次开始，这样势必减少了大量的结点数。本文以圆盘上的积分为例给出了详细的构造过程，并且得到了许多相应的求积公式，有些公式的结点数已经达到最小。最后，本文还给出了不同权系数下直交多项式之间的一个关系式。
(7)借助多项式理想的相关知识，本文研究了一个事先给定结点的数值积分公式的构造方法。把给定的结点限制为一个低次求积公式的结点，由此进一步得到一个更高次代数精精度的求积公式。如此可得到一个嵌入列，即低次求积公式的结点都是高次积分公式的结点。此类研究的意义在于：多元数值积分的误差分析一直是一个难题，人们通常以两个不同精度的求积公式的差表示积分误差。此时，利用嵌入式积分公式序列将有利于分析和减少计算量。另外，文中还给出了一个二元非插值型积分公式的例子。
Orthogonal polynomials and numerical integration in several variables are two hot problems for discussion in the field of numerical approximation, and have many important applications in computational geometry, scientific computing, harmonic analysis, the theory of special functions, probability and statistics and so on. Furthermore both of them are also closely related. In this paper, some problems about these two topics are discussed, which mainly include some extensions of general properties of one dimensional orthogonal polynomials and the construction of cubature formulae with special structures.
(1) Historically, the orthogonal polynomials originated in the theory of continued fractions. But continued fractions have been gradually abandoned as a starting point for the theory of orthogonal polynomials. One of the important reasons is that it is difficult to be extended. To overcome it, professor Luo Zhongxuan and professor Wang Renhong introduced the concepts of invariant factor for orthogonal polynomials of two variables. By invariant factor, they proved the Stieltjes theorem of two variables, in which invariant factors play an important role. In this paper, some further properties of invariant factor are discussed, especially the properties of its zeros. The following results are presented: Every zero of invariant factor must be an eigenvalue of the corresponding truncated Jacobi matrix, and hence real. Especially, if all the zeros of invariant factor are distinct from each other, then the invariant factor equals to the characteristic polynomial of the corresponding truncated Jacobi matrix up to a constant multiple. Moreover, for every zero of invariant factor, there must exist an orthogonal polynomial factored into two polynomials, from which an zero location of invariant factor, similar to the case of one dimensional orthogonal polynomial, can be obtained.
(2) The properties of zeros for one dimensional orthogonal polynomial are the important part of the general properties. Usually to generalize the corresponding properties, common zeros of orthogonal polynomials are used in the case of several variables. Thus part of theory in one variable can be extended to several variables. But the common zeros location remains unknown. In one variable, all the zeros of orthogonal polynomials are located in the interior of integration interval, which is different from the case of two variables. The common zeros of all multivariate orthogonal polynomials with same total degree can lie outside of the region. An obvious example is the orthogonal polynomials on the two dimensional spherical shell. The original point is the common zero of all orthogonal polynomials with odd degree but lie outside of the spherical shell. Base on the properties of invariant factor we prove that all the common zeros lie in the interior of the convex closed hull of the integration region.
(3) Numerical integration is closely related to orthogonal polynomials, and in one variable it usually takes the zeros of orthogonal polynomials as its knots which benefit from the good properties of univariate orthogonal polynomials. However, in higher dimension we know a litter about this. One of the difficulties is one does not know the exact number of common zeros for a set of polynomials. It follows from the property of invariant factor that for every zero of invariant factor there must exist an orthogonal polynomial which can be factored into a product of two polynomials of lower degree. Especially, for n=2, these two polynomials of lower degree is of degree 1. Stroud proved that if two orthogonal polynomials have four intersections, then these points must constitute a set of knots of a cubature formula of degree 3. Obviously, it is easy to compute the number of the common zeros of two orthogonal polynomials with the factoring property. It is based on this idea that we present a method to construct the cubature formula of degree 3 with 4 knots.
(4) To investigate the properties of multivariate orthogonal polynomials, we extend the concept of invariant factor and get the similar results with two variables. The presented results include multivariate Stieltjes type theorem, the relations among invariant factor, multivariate orthogonal polynomials and Jacobi matrix, and common zeros location of multivariate orthogonal polynomials and so on.
(5) Among various methods of constructing cubature formulae, the method of Cartesian product stands out with its simplicity. Briefly, the method for constructing product formulas is the method of separation of variables. If we can find a (nonlinear) transform which transforms the monomial integral into the product of d single integrals and if suitable formulas are known for these single integrals, then one can combine these formulas to give a formula. The most undesired property of product formulas is that the number of points increases very rapidly as d increases. To avoid it partly, we present a proper transformation which transforms the initial integration into the product of single integrations and reduces the number of knots greatly. Furthermore, some knots are used repeatedly by the cubature formulae of any degree.
(6) Polynomial interpolation and quadrature formula are closely related. To investigate the interpolation problem of two variables, Liang presented a kind of recursive method of constructing properly posed set of nodes. This paper consider the corre- sponding recursive method of construction of numericai integration. First we present a construction method of cubature formulae with some knots along the selected algebraic curve. By selecting different algebraic curves and repeating the above process, we will obtain a cubature formula and then a recursive method. To reduce the number of the knots, we take the polynomial corresponding to the selected curve as an orthogonal polynomial. Thus the number of the knots of final cubature formula will be reduced greatly. Taking the unit disk as an example, a detailed construction process is given. Some cubature formula have gotten to the lower bound.
(7) By the knowledge of polynomial ideal, we present a construction method of cubature formula with some fixed points. If we take the fixed points as the nodes of a cubature of lower degree and get a new cubature of higher degree, then an embedded formula sequence is obtained. This kind of embedded sequence can be used to estimate the errors. Furthermore, a noninterpolationary cubature formula is presented.
引文
[1] Luo Z X, Wang R H. Stieltjes type theorems for orthogonal polynomials of two variables. Journal of Mathematical Analysis and Applications, 2002, 268(1): 171-183.
[2] Stroud A H. Integration formulas and orthogonal polynomials for two variables. SIAM J. Numer. Anal., 1969, 6: 222-229.
[3] Moller H M. Polynomideale und kubaturformeln: [Disseration]. Universitat Dort-mund, 1973.
[4] Moller H M. An inverse problem for cubature formulae. Computational Technologies, 2004, 9(7): 13-20.
[5] 梁学章．关于多元函数的插值与逼近：(硕士学位论文)．长春：吉林大学，1965．
[6] 梁学章，崔利宏．代数曲线上的lagrange插值．吉林大学自然科学学报，2001，39(3)：17-20．
[7] 梁学章．二元插值的适定结点组与迭加插值．吉林大学自然科学学报，1979，17(1)：27-32．
[8] 王仁宏，梁学章．多元函数逼近．北京：科学出版社，1988．
[9] 崔利宏．多元Lagrange插值与多元Kergin插值：(博士学位论文)．长春：吉林大学，2003．
[10] Szego G. Orthogonal Polynomials, vol. 23. Providence, RI: Amer. Math. Soc. Colloq. PUbl., 1975, 4th ed.
[11] Freud G. Orthogonal Polynomials. New York: Pergamon, 1966.
[12] Nevai P. Orthogonal Polynomials, vol. 18. Mem. Amer. Math. Soc., 1979.
[13] Appell P, Kampe de Feriet J. Fonctions Hypergeometriques et Hyperspheriques-Polynomes d' Hermite. Paris: Gauthier Villars, 1926.
[14] Erdelyi A, Magnus W, Oberhettinger F, Tricomi F G. Higher transcendental functions. New York: McGraw-Hill, 1953.
[15] Kowalski M A. The recursion formulas for orthogonal polynomials in n variables. SIAM J. Math. Anal., 1982, 13(2): 309-315.
[16] Kowalski M A. Orthogonality and recursion formulas for polynomials in n variables. SIAM J. Math. Anal., 1982, 13(2): 316-323.
[17] Dunkl C F, Xu Y. Orthogonal Polynomials of Several Variables. Cambridge: Cambridge University Press, 2001.
[18] Luo Z X, Meng Z L, Liu F S. Multivariate stieltjes type theorems and common zeros location of multivariate orthogonal polynomials, submitted for publication.
[19] Walker R J.. Algebraic Curves. New York: Springer-Verlag, 1978.
[20] Cox D A, Little J B, O'Shea D. Ideals, Varieties, and Algorithms. Berlin: Springer, 1996, second ed.
[21] Cox D A, Little J B, O'Shea D. Using algebraic geometry. New York: Springer, 1998.
[22] Mysovskikh I P. Interpolatory cubature formulas. Moscow: "Nauka", 1981. (Rus-sian).
[23] Dunkl C F. Differential-difference operators associated to reflection groups. Trans. Amer. Math. Soc., 1989, 311: 167-183.
[24] Xu Y. Monomial orthogonal polynomials of several variables. J. Approx. Theory, 2005, 133: 1-37.
[25] Mysovskikh I P. A representation of the reproducing kernel of a sphere. Comp. Math. Math. Phys., 1996, 36: 303-308.
[26] Xu Y. Asymptotics for orthogonal polynomials and Christoffel functions on a ball. Methods Appl. Anal., 1996, 3: 257-272.
[27] Mysovskikh I P. Numerical charicteristics of orthogonal polynomials in two variables. Vestnik Leninggrad Univ. Math., 1976, (3): 323-332.
[28] 周恒．多元正交多项式的理论与应用研究：(博士学位论文)．大连：大连理工大学，2004．
[29] Engels H. Numerical quadrature and cubature. New York: Academic Press, 1980.
[30] Rabinowitz P, Richter N. Perfectly symmetric two-dimensional integration formulas with minimal number of points. Math. Comp., 1969, 23: 765-799.
[31] Maxwell J C. On approximation multiple integration between limits of summation. Proc. Cambridge Philos. Soc., 1877, 3: 39-47.
[32] Appell P. Sur une classe de polynome a deux variables et le calcul approche des integrales double. Ann. Fac. Sci. Univ. Toulouse, 1890, 4: H1-H20.
[33] Radon J. Zur mechanischen Kubatur. Monatsh. Math., 1948, 52: 286-300.
[34] Cools R. A survey of methods for constructing cubature formulae, in: Espelid T O, Genz A, eds., Numerical Integration Recent Developments, Software and Applications. Dordrecht: Kluwer Academic Publishers, 1992: 1-24.
[35] Cools R. Constructing cubature formulae: The science behind the art. Acta Numerica, vol. 6. Cambridge: Cambridge University Press, 1997: 1-54.
[36] Cools R. Advances in multidimensional integration. Journal of Computational and Applied Mathematics, 2002, 149: 1-12.
[37] Stroud A H. Approximate Calculation of Multiple Integrals. N. J.: Prentice-Hall, Englewood Cliffs, 1971.
[38] Moller H M. Kubaturformeln mit minimaler Knotenzahl. Numer. Math., 1976, 25: 185-200.
[39] Morrow C R, Patterson T N L. Construction of algebraic cubature rules using polynomial ideal theory. SIAM J. Numer. Anal., 1978, 15: 953-976.
[40] Godzina G, Erlangen. Blending methods for two classical integrals. Computing, 1995, 54: 273-282.
[41] Xu Y. Lagrange interpolation on chebyshev points of two variables. J. Approx. Theory, 1996, 87: 220-238.
[42] Cools R, Schmid H J. On the (non)-existence of some cubature formulas: gaps between a theory and its applications. Journal of Complexity, 2003, 19(3): 403-405.
[43] 华罗庚，王元．数值积分及其应用．北京：科学出版社，1963．
[44] 徐利治，周蕴时．高维数值积分．北京：科学出版社，1980．
[45] Cools R. An encyclopaedia of cubature formulas. Journal of Complexity, 2003, 19: 445-453.
[46] Cools R, Rabinowitz P. Monomial cubature rules since Stroud: a compilation. J. Comput. Appl. Math., 1993, 48: 309-326.
[47] Cools R. Monomial cubature rules since "Stroud": a compilation-part ii. J. Comput. Appl. Math., 1999, 112: 21-27.
[48] Schmid H J. On cubature formulae with a minimal number of knots. Numer. Math., 1978, 31: 281-297.
[49] Schmid H J. Two-dimensional minimal cubature formulas and matrix equations. SIAM J. Matrix Anal. Appl., 1995, 16: 898-921.
[50] Moller H M. The construction of cubature formulae and ideals of principal classes, in: Hammerlin G, ed., Numerical Integration, International Series of Numerical Mathematics, vol. 45. Basel: Birkhauser, 1979: 221-230.
[51] Cools R, Sloan I H. Minimal cubature formulae of trigonometric degree. Math. Comp., 1996, 65(216): 1583-1600.
[52] Mysovskikh I P. Cubature formulae that are exact for trigonometric polynomials. Report TW 324, Department of Computer Science, K. U. Leuven, 2001.
[53] Cools R, Santos-Leon J C. Cubature formulas of a nonalgebraic degree of precision. Report TW 301, Department of Computer Science, K. U. Leuven, 2000.
[54] Davis P J, Rabinowitz P. Methods of Numerical Integration. New York: Academic Press, 1975.
[55] Lobatto R. Lessen over de Integral-Rekening. The Hague, 1852.
[56] Stancu D D, H S A. Quadrature formulas with simple gaussian nodes and multiple fixed nodes. Math. Comp., 1963, 17: 384-394.
[57] 柳重堪．正交函数及其应用．北京：国防工业出版社，1982．
[58] Natonson. Konstruktive Funktionentheorie. Berlin, 1955.
[59] 王仁宏．数值逼近．北京：高等教育出版社，1999．
[60] 徐利治，周蕴时，何天晓．高维数值积分选讲．合肥：安徽教育出版社，1985．
[61] Brezinski C. Pade-Type Approximation and General Orthogonal Polynomials. Basel: Birkhauser, 1980.
[62] Rockafellar R T. Convex Analysis. N. J.: Princeton University Press, 1970.
[63] Luo Z X, Meng Z L. On the properties of bivariate orthogonal polynomials. J. Information and Computing Sciences, 2004, 1(1): 103-106.
[64] Luo Z X, Meng Z L. A result on common zeros location of orthogonal polynomials in two variables. Journal of Mathematical Analysis and Applications, 2006, 324(2): 785-789.
[65] Gunther G. Third degree integration formulas with four real points and positive weights in two dimensions. SIAM J. Numer. Anal., 1974, 11(3): 480-493.
[66] Luo Z X, Meng Z L. The invariant factors of orthogonal polynomial for two variables and numerical integration formulas. Chinese Journal of Numerical Mathematics and Application, 2005, 27(3): 79-91.
[67] 罗钟铉，孟兆良．二元直交多项式的不变因子与数值积分公式．计算数学，2005，27(2)：199-208．
[68] Stroud A H, Secrest D. Approximate integration formulas for certain spherically symmetric regions. Math. Comp., 1963, 17: 105-135.
[69] Stroud A H, Secrest D. Gaussian Quadrature Formulas. N. J.: Prentice-Hall, Englewood Cliffs, 1966.
[70] Luo Z X, Meng Z L. Cubature formulas over n-sphere. Journal of Computational and Applied Mathematics, to appear.
[71] Xu Y. Polynomial interpolation in several variables, cubature formulae, and ideals. Advances in Comp. Math., 2000, 12: 363-376.
[72] Liang. X Z, Lu C M. Properly posed set of nodes for bivariate lagrange interpolation. Chui C K, in: Chui Charles K L, Schumaker L, eds., Approximation Theory Ⅸ, vol. 1. 1998: 189-196.
[73] 张洁琳．n元函数的Lagrange插值与二维数字图像的小波逼近：(博士学位论文)．长春：吉林大学．2003．
[74] Morrow C R, Patterson T N L. The construction of algebraic cubature formulae by the distribution of nodes along selected lines. SIAM Journal on Numerical Analysis, 1985, 22(6): 1178-1190.
[75] Cools R, Kim K. A survey of known and new cubature formulas for the unit disk. Korean J. Comp. Appl. Math., 2000, 7(3): 477-485.
[76] Cools R, Haegemans A. Construction of fully symmetric cubature formulae of degree 4k-3 for fully symmetric planar regions. Report TW 71, Department of Computer Science, K. U. Leuven, 1985.
[77] Cools R, Haegemans A. Automatic computation of knots and weights of cubature formula for circular symmetric planar regions. Report TW 77, Department of Computer Science, K. U. Leuven, 1986.
[78] Kim K, Song M. Symmetric formulas over a unit disk. Korean J. Comp. Appl. Math., 1997, 4(1): 179-192.
[79] Meng Z L, Luo Z X. The construction of cubature formulae with some knots along the selected algebraic curve. J. Information and Computational Science, 2005, 2(3): 539-546.
[80] Kronrod A S. Nodes and weights of quadrature formulas. New York: Consultants Bureau, 1965.
[81] Guessab A. Numerical cubature formulae with preassigned knots. Numer. Math., 1987, 52: 467-478.
[82] Xu Y. Constructing cubature formulae by the method of reproducing kernel. Numer. Math., 2000, 85:155-173.
[83] Rasputin G G. The construction of cubature formulas containing previously specified nodes. Soviet Math., 1986, 30:58-67.
[84] Rasputin G G. On the construction of cubature formulas containing preassigned knots. Metody Vychisl, 1983, 13:122-128.
[85] Cools R, Haegemans A. Optimal addition of knots to cubature formulas for planar regions. Numer. Math., 1986, 49:269-274.
[86] Cools R, Haegemans A. Construction of sequences of embedded cubature formulae for circular symmetric planar regions. in: Keast P, Fairweather P, eds., Numerical Integration (Dordrecht). Reidel Publ. Comp., 1987:165-172.
[87] Cools R, Haegemans A. An embedded pair of cubature formulae of degree 5 and 7 for the triangle. BIT, 1988, 28:357-359.
[88] Cools R, Haegemans A. On the construction of multi-dimensional embedded cubature formulae. Numer. Math., 1989, 55:735-745.
[89] Luo Z X, Meng Z L, Liu F S. The construction of cubature formulae with preassigned nodes, submitted for publication.
[2] Stroud A H. Integration formulas and orthogonal polynomials for two variables. SIAM J. Numer. Anal., 1969, 6: 222-229.
[3] Moller H M. Polynomideale und kubaturformeln: [Disseration]. Universitat Dort-mund, 1973.
[4] Moller H M. An inverse problem for cubature formulae. Computational Technologies, 2004, 9(7): 13-20.
[5] 梁学章．关于多元函数的插值与逼近：(硕士学位论文)．长春：吉林大学，1965．
[6] 梁学章，崔利宏．代数曲线上的lagrange插值．吉林大学自然科学学报，2001，39(3)：17-20．
[7] 梁学章．二元插值的适定结点组与迭加插值．吉林大学自然科学学报，1979，17(1)：27-32．
[8] 王仁宏，梁学章．多元函数逼近．北京：科学出版社，1988．
[9] 崔利宏．多元Lagrange插值与多元Kergin插值：(博士学位论文)．长春：吉林大学，2003．
[10] Szego G. Orthogonal Polynomials, vol. 23. Providence, RI: Amer. Math. Soc. Colloq. PUbl., 1975, 4th ed.
[11] Freud G. Orthogonal Polynomials. New York: Pergamon, 1966.
[12] Nevai P. Orthogonal Polynomials, vol. 18. Mem. Amer. Math. Soc., 1979.
[13] Appell P, Kampe de Feriet J. Fonctions Hypergeometriques et Hyperspheriques-Polynomes d' Hermite. Paris: Gauthier Villars, 1926.
[14] Erdelyi A, Magnus W, Oberhettinger F, Tricomi F G. Higher transcendental functions. New York: McGraw-Hill, 1953.
[15] Kowalski M A. The recursion formulas for orthogonal polynomials in n variables. SIAM J. Math. Anal., 1982, 13(2): 309-315.
[16] Kowalski M A. Orthogonality and recursion formulas for polynomials in n variables. SIAM J. Math. Anal., 1982, 13(2): 316-323.
[17] Dunkl C F, Xu Y. Orthogonal Polynomials of Several Variables. Cambridge: Cambridge University Press, 2001.
[18] Luo Z X, Meng Z L, Liu F S. Multivariate stieltjes type theorems and common zeros location of multivariate orthogonal polynomials, submitted for publication.
[19] Walker R J.. Algebraic Curves. New York: Springer-Verlag, 1978.
[20] Cox D A, Little J B, O'Shea D. Ideals, Varieties, and Algorithms. Berlin: Springer, 1996, second ed.
[21] Cox D A, Little J B, O'Shea D. Using algebraic geometry. New York: Springer, 1998.
[22] Mysovskikh I P. Interpolatory cubature formulas. Moscow: "Nauka", 1981. (Rus-sian).
[23] Dunkl C F. Differential-difference operators associated to reflection groups. Trans. Amer. Math. Soc., 1989, 311: 167-183.
[24] Xu Y. Monomial orthogonal polynomials of several variables. J. Approx. Theory, 2005, 133: 1-37.
[25] Mysovskikh I P. A representation of the reproducing kernel of a sphere. Comp. Math. Math. Phys., 1996, 36: 303-308.
[26] Xu Y. Asymptotics for orthogonal polynomials and Christoffel functions on a ball. Methods Appl. Anal., 1996, 3: 257-272.
[27] Mysovskikh I P. Numerical charicteristics of orthogonal polynomials in two variables. Vestnik Leninggrad Univ. Math., 1976, (3): 323-332.
[28] 周恒．多元正交多项式的理论与应用研究：(博士学位论文)．大连：大连理工大学，2004．
[29] Engels H. Numerical quadrature and cubature. New York: Academic Press, 1980.
[30] Rabinowitz P, Richter N. Perfectly symmetric two-dimensional integration formulas with minimal number of points. Math. Comp., 1969, 23: 765-799.
[31] Maxwell J C. On approximation multiple integration between limits of summation. Proc. Cambridge Philos. Soc., 1877, 3: 39-47.
[32] Appell P. Sur une classe de polynome a deux variables et le calcul approche des integrales double. Ann. Fac. Sci. Univ. Toulouse, 1890, 4: H1-H20.
[33] Radon J. Zur mechanischen Kubatur. Monatsh. Math., 1948, 52: 286-300.
[34] Cools R. A survey of methods for constructing cubature formulae, in: Espelid T O, Genz A, eds., Numerical Integration Recent Developments, Software and Applications. Dordrecht: Kluwer Academic Publishers, 1992: 1-24.
[35] Cools R. Constructing cubature formulae: The science behind the art. Acta Numerica, vol. 6. Cambridge: Cambridge University Press, 1997: 1-54.
[36] Cools R. Advances in multidimensional integration. Journal of Computational and Applied Mathematics, 2002, 149: 1-12.
[37] Stroud A H. Approximate Calculation of Multiple Integrals. N. J.: Prentice-Hall, Englewood Cliffs, 1971.
[38] Moller H M. Kubaturformeln mit minimaler Knotenzahl. Numer. Math., 1976, 25: 185-200.
[39] Morrow C R, Patterson T N L. Construction of algebraic cubature rules using polynomial ideal theory. SIAM J. Numer. Anal., 1978, 15: 953-976.
[40] Godzina G, Erlangen. Blending methods for two classical integrals. Computing, 1995, 54: 273-282.
[41] Xu Y. Lagrange interpolation on chebyshev points of two variables. J. Approx. Theory, 1996, 87: 220-238.
[42] Cools R, Schmid H J. On the (non)-existence of some cubature formulas: gaps between a theory and its applications. Journal of Complexity, 2003, 19(3): 403-405.
[43] 华罗庚，王元．数值积分及其应用．北京：科学出版社，1963．
[44] 徐利治，周蕴时．高维数值积分．北京：科学出版社，1980．
[45] Cools R. An encyclopaedia of cubature formulas. Journal of Complexity, 2003, 19: 445-453.
[46] Cools R, Rabinowitz P. Monomial cubature rules since Stroud: a compilation. J. Comput. Appl. Math., 1993, 48: 309-326.
[47] Cools R. Monomial cubature rules since "Stroud": a compilation-part ii. J. Comput. Appl. Math., 1999, 112: 21-27.
[48] Schmid H J. On cubature formulae with a minimal number of knots. Numer. Math., 1978, 31: 281-297.
[49] Schmid H J. Two-dimensional minimal cubature formulas and matrix equations. SIAM J. Matrix Anal. Appl., 1995, 16: 898-921.
[50] Moller H M. The construction of cubature formulae and ideals of principal classes, in: Hammerlin G, ed., Numerical Integration, International Series of Numerical Mathematics, vol. 45. Basel: Birkhauser, 1979: 221-230.
[51] Cools R, Sloan I H. Minimal cubature formulae of trigonometric degree. Math. Comp., 1996, 65(216): 1583-1600.
[52] Mysovskikh I P. Cubature formulae that are exact for trigonometric polynomials. Report TW 324, Department of Computer Science, K. U. Leuven, 2001.
[53] Cools R, Santos-Leon J C. Cubature formulas of a nonalgebraic degree of precision. Report TW 301, Department of Computer Science, K. U. Leuven, 2000.
[54] Davis P J, Rabinowitz P. Methods of Numerical Integration. New York: Academic Press, 1975.
[55] Lobatto R. Lessen over de Integral-Rekening. The Hague, 1852.
[56] Stancu D D, H S A. Quadrature formulas with simple gaussian nodes and multiple fixed nodes. Math. Comp., 1963, 17: 384-394.
[57] 柳重堪．正交函数及其应用．北京：国防工业出版社，1982．
[58] Natonson. Konstruktive Funktionentheorie. Berlin, 1955.
[59] 王仁宏．数值逼近．北京：高等教育出版社，1999．
[60] 徐利治，周蕴时，何天晓．高维数值积分选讲．合肥：安徽教育出版社，1985．
[61] Brezinski C. Pade-Type Approximation and General Orthogonal Polynomials. Basel: Birkhauser, 1980.
[62] Rockafellar R T. Convex Analysis. N. J.: Princeton University Press, 1970.
[63] Luo Z X, Meng Z L. On the properties of bivariate orthogonal polynomials. J. Information and Computing Sciences, 2004, 1(1): 103-106.
[64] Luo Z X, Meng Z L. A result on common zeros location of orthogonal polynomials in two variables. Journal of Mathematical Analysis and Applications, 2006, 324(2): 785-789.
[65] Gunther G. Third degree integration formulas with four real points and positive weights in two dimensions. SIAM J. Numer. Anal., 1974, 11(3): 480-493.
[66] Luo Z X, Meng Z L. The invariant factors of orthogonal polynomial for two variables and numerical integration formulas. Chinese Journal of Numerical Mathematics and Application, 2005, 27(3): 79-91.
[67] 罗钟铉，孟兆良．二元直交多项式的不变因子与数值积分公式．计算数学，2005，27(2)：199-208．
[68] Stroud A H, Secrest D. Approximate integration formulas for certain spherically symmetric regions. Math. Comp., 1963, 17: 105-135.
[69] Stroud A H, Secrest D. Gaussian Quadrature Formulas. N. J.: Prentice-Hall, Englewood Cliffs, 1966.
[70] Luo Z X, Meng Z L. Cubature formulas over n-sphere. Journal of Computational and Applied Mathematics, to appear.
[71] Xu Y. Polynomial interpolation in several variables, cubature formulae, and ideals. Advances in Comp. Math., 2000, 12: 363-376.
[72] Liang. X Z, Lu C M. Properly posed set of nodes for bivariate lagrange interpolation. Chui C K, in: Chui Charles K L, Schumaker L, eds., Approximation Theory Ⅸ, vol. 1. 1998: 189-196.
[73] 张洁琳．n元函数的Lagrange插值与二维数字图像的小波逼近：(博士学位论文)．长春：吉林大学．2003．
[74] Morrow C R, Patterson T N L. The construction of algebraic cubature formulae by the distribution of nodes along selected lines. SIAM Journal on Numerical Analysis, 1985, 22(6): 1178-1190.
[75] Cools R, Kim K. A survey of known and new cubature formulas for the unit disk. Korean J. Comp. Appl. Math., 2000, 7(3): 477-485.
[76] Cools R, Haegemans A. Construction of fully symmetric cubature formulae of degree 4k-3 for fully symmetric planar regions. Report TW 71, Department of Computer Science, K. U. Leuven, 1985.
[77] Cools R, Haegemans A. Automatic computation of knots and weights of cubature formula for circular symmetric planar regions. Report TW 77, Department of Computer Science, K. U. Leuven, 1986.
[78] Kim K, Song M. Symmetric formulas over a unit disk. Korean J. Comp. Appl. Math., 1997, 4(1): 179-192.
[79] Meng Z L, Luo Z X. The construction of cubature formulae with some knots along the selected algebraic curve. J. Information and Computational Science, 2005, 2(3): 539-546.
[80] Kronrod A S. Nodes and weights of quadrature formulas. New York: Consultants Bureau, 1965.
[81] Guessab A. Numerical cubature formulae with preassigned knots. Numer. Math., 1987, 52: 467-478.
[82] Xu Y. Constructing cubature formulae by the method of reproducing kernel. Numer. Math., 2000, 85:155-173.
[83] Rasputin G G. The construction of cubature formulas containing previously specified nodes. Soviet Math., 1986, 30:58-67.
[84] Rasputin G G. On the construction of cubature formulas containing preassigned knots. Metody Vychisl, 1983, 13:122-128.
[85] Cools R, Haegemans A. Optimal addition of knots to cubature formulas for planar regions. Numer. Math., 1986, 49:269-274.
[86] Cools R, Haegemans A. Construction of sequences of embedded cubature formulae for circular symmetric planar regions. in: Keast P, Fairweather P, eds., Numerical Integration (Dordrecht). Reidel Publ. Comp., 1987:165-172.
[87] Cools R, Haegemans A. An embedded pair of cubature formulae of degree 5 and 7 for the triangle. BIT, 1988, 28:357-359.
[88] Cools R, Haegemans A. On the construction of multi-dimensional embedded cubature formulae. Numer. Math., 1989, 55:735-745.
[89] Luo Z X, Meng Z L, Liu F S. The construction of cubature formulae with preassigned nodes, submitted for publication.