多元分次Lagrange插值
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摘要
本文对多元多项式分次插值适定结点组的构造理论进行了深入的研究与探讨。在沿无重复分量代数曲线进行Lagrange插值的基础上，我们给出了沿无重复分量分次代数曲线进行分次Lagrange插值的方法，并利用这一结果进一步给出在R~2上构造分次Lagrange插值适定结点组的基本方法。另外，利用弱Grobner基这一新的数学概念，以及构造平面代数曲线上插值适定结点组的理论，我们进一步给出了构造平面分次代数曲线上分次插值适定结点组的方法。同时，利用代数几何学中关于理想和代数簇的理论，研究了代数超曲面上分次插值适定结点组的几何结构，通过上述理论的研究，并利用无重复分量代数超曲面上的分次插值适定结点组的构造方法，我们又得到了构造高维空间中分次插值适定结点组的递归构造方法，从而基本上弄清了多元分次Lagrange插值适定结点组的几何结构和基本特征。另一方面，在研究插值结点组的移动对插值适定性的影响这一问题时，我们对以往一个重要定理加以推广并给出了证明，从而得到了更一般性的结论。
The constitution theory of a properly posed set of nodes forthe multivariate polynomial graded interpolation is studied deeply inthe paper. On the basis of Lagrange interpolation which along the al-gebraic curve without multiple factors,we give the approach of gradedLagrange interpolation which along the algebraic curve without mul-tiple factors. Futhermore,using this result we give a basic method toconstruct the graded Lagrange interpolation in R~2. In addition,usingweak Grobner basis' method which is a new mathematic concept andthe theory for constructing the properly posed set of nodes for inter-polation on plane algebraic curve,we give the method to construct theproperly posed set of nodes for graded interpolation on plane algebraiccurve accordingly. At the same time, using the results of algebraic vari-ety and ideal in algebraic geometry, we study the geometrical structureof properly posed set of nodes for graded interpolation on algebraic hy-persurface, more over,we give a Hyperplane Superposition Process toconstruct the properly posed set of nodes for graded interpolation onalgebraic hypersurface,therefore we make clear the geometrical struc-ture of properly posed set of nodes for multivariate graded interpola-tion basically.On the other hand,when we study the effect of movinginterpolation nodes to the property for interpolation,we improve andprove an important theorem gao, thus we acquire the general conclu-sion.
The constitution theory of a properly posed set of nodes forthe multivariate polynomial graded interpolation is studied deeply inthe paper. On the basis of Lagrange interpolation which along the al-gebraic curve without multiple factors,we give the approach of gradedLagrange interpolation which along the algebraic curve without mul-tiple factors. Futhermore,using this result we give a basic method toconstruct the graded Lagrange interpolation in R~2. In addition,usingweak Grobner basis' method which is a new mathematic concept andthe theory for constructing the properly posed set of nodes for inter-polation on plane algebraic curve,we give the method to construct theproperly posed set of nodes for graded interpolation on plane algebraiccurve accordingly. At the same time, using the results of algebraic vari-ety and ideal in algebraic geometry, we study the geometrical structureof properly posed set of nodes for graded interpolation on algebraic hy-persurface, more over,we give a Hyperplane Superposition Process toconstruct the properly posed set of nodes for graded interpolation onalgebraic hypersurface,therefore we make clear the geometrical struc-ture of properly posed set of nodes for multivariate graded interpola-tion basically.On the other hand,when we study the effect of movinginterpolation nodes to the property for interpolation,we improve andprove an important theorem gao, thus we acquire the general conclu-sion.
引文
[1] 梁学章，关于多元函数的插值与逼近，吉林大学研究生毕业论文，长春，1965．
[2] 崔利宏，多元Lagrange插值与多元Kergin插值，吉林大学博士学位论文，2003．
[3] X.Z.Liang,C.M.Lu, Properly posed set of nodes for bivariate interpolation, Approximation Theory Ⅸ, Vol.2,Computation Aspect, Vanderbit University Press, 1998,189-196.
[4] Ward Cheney, Will Light, A Course in Approximation Theory，机械工业出版社，2004．
[5] X.Z.Liang,C.M.Lu and R.Z.Feng, Properly posed sets of nodes for multivariate Lagrange interpolation in C~s, SIAM, Numer. Anal, 2001,NO.2 Vol.39,578-595.
[6] J. G. Semple and L. Roth, Introduction to Algebraic Geometry, Oxford at the Clarendon Press,1949.
[7] David Cox, John Little, Donal O'sha, Ideals, Varieties and Algorithms, Springer-Verlag, NewYork. 1992.
[8] M.G.Marinari,H. M. Moeller, T. Mora, Grobner bases of ideas given by dual bases, in processings of ISSAC,1991.
[9] 任红菊，超限插值与曲面拼接技术，吉林大学博士学位论文，1999．
[10] 吕春梅，多元多项式插值，吉林大学博士学位论文，1997．
[11] 梁学章，关于多元函数的插值与逼近，高等学校计算数学学报，1979，Vol．1，No．1，123-124．
[12] 王仁宏，梁学章，多元函数逼近，科学出版社，1988．
[13] 周蕴时，苏志勋，奚涌江等，CAGD中的曲线和曲面，吉林大学出版社，1993．
[14] 孙永生，函数逼近论，北京师范大学出版社，1989．
[15] P．格列菲斯，代数曲线，北京大学出版社，1985．
[16] 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，No．3，17-20．
[17] K.C.Chung and T.H.Yao, On lattices admitting unique Lagrange interpolation, SIAM J. Numer,Anal, 1977,No. 14,735-741.
[18] M. Gasca and J. I. Maeztu, On Lagrange and Hermite interpolation in R~k, Numer Math, 1982,Vol.39,1-14.
[19] R.A.Lorentz,Multivariate Hermite interpolation by algebraic polynomials: A survey, Journal of Computational and Applide Mathematics,200,Vol.122,167-202.
[20] 朱平，傅凯新，十字型结点组及R~2上的插值，高等学校计算数学学报，1995，Vol．1．
[21] 朱平，竖线型结点组上的插值及向高维情形的推广，数学杂志，1998，Vol．18，No．4．
[22] De. Boor and A. Ron, The least solution for the polynomial interpolation problem, Math.Z,1992,Vol.210,347-378.
[23] M. Gasca and T. Sauer, Polynomial interpolation in several variables, Advances in Computational Mathematics,2000,Vol. 12,377-409.
[2] 崔利宏，多元Lagrange插值与多元Kergin插值，吉林大学博士学位论文，2003．
[3] X.Z.Liang,C.M.Lu, Properly posed set of nodes for bivariate interpolation, Approximation Theory Ⅸ, Vol.2,Computation Aspect, Vanderbit University Press, 1998,189-196.
[4] Ward Cheney, Will Light, A Course in Approximation Theory，机械工业出版社，2004．
[5] X.Z.Liang,C.M.Lu and R.Z.Feng, Properly posed sets of nodes for multivariate Lagrange interpolation in C~s, SIAM, Numer. Anal, 2001,NO.2 Vol.39,578-595.
[6] J. G. Semple and L. Roth, Introduction to Algebraic Geometry, Oxford at the Clarendon Press,1949.
[7] David Cox, John Little, Donal O'sha, Ideals, Varieties and Algorithms, Springer-Verlag, NewYork. 1992.
[8] M.G.Marinari,H. M. Moeller, T. Mora, Grobner bases of ideas given by dual bases, in processings of ISSAC,1991.
[9] 任红菊，超限插值与曲面拼接技术，吉林大学博士学位论文，1999．
[10] 吕春梅，多元多项式插值，吉林大学博士学位论文，1997．
[11] 梁学章，关于多元函数的插值与逼近，高等学校计算数学学报，1979，Vol．1，No．1，123-124．
[12] 王仁宏，梁学章，多元函数逼近，科学出版社，1988．
[13] 周蕴时，苏志勋，奚涌江等，CAGD中的曲线和曲面，吉林大学出版社，1993．
[14] 孙永生，函数逼近论，北京师范大学出版社，1989．
[15] P．格列菲斯，代数曲线，北京大学出版社，1985．
[16] 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，No．3，17-20．
[17] K.C.Chung and T.H.Yao, On lattices admitting unique Lagrange interpolation, SIAM J. Numer,Anal, 1977,No. 14,735-741.
[18] M. Gasca and J. I. Maeztu, On Lagrange and Hermite interpolation in R~k, Numer Math, 1982,Vol.39,1-14.
[19] R.A.Lorentz,Multivariate Hermite interpolation by algebraic polynomials: A survey, Journal of Computational and Applide Mathematics,200,Vol.122,167-202.
[20] 朱平，傅凯新，十字型结点组及R~2上的插值，高等学校计算数学学报，1995，Vol．1．
[21] 朱平，竖线型结点组上的插值及向高维情形的推广，数学杂志，1998，Vol．18，No．4．
[22] De. Boor and A. Ron, The least solution for the polynomial interpolation problem, Math.Z,1992,Vol.210,347-378.
[23] M. Gasca and T. Sauer, Polynomial interpolation in several variables, Advances in Computational Mathematics,2000,Vol. 12,377-409.