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The Discrete Approximation Problem for a Special Case of Hermite-Type Polynomial Interpolation

GONG Yihe1, JIANG Xue2, ZHANG Shugong3   

  1. 1. College of Science, Northeast Electric Power University, Jilin 132000, China;
    2. School of Mathematics and Systematic Sciences, Shenyang Normal University, Shenyang 110034, China;
    3. Institute of Mathematics, Key Lab. of Symbolic Computation and Knowledge Engineering (Ministry of Education), Jilin University, Changchun 130012, China
  • Received:2021-03-12 Revised:2021-05-17 Online:2022-10-25 Published:2022-10-12
  • Supported by:
    This paper was supported by the National Natural Science Foundation of China under Grant Nos.11901402 and 11671169.

GONG Yihe, JIANG Xue, ZHANG Shugong. The Discrete Approximation Problem for a Special Case of Hermite-Type Polynomial Interpolation[J]. Journal of Systems Science and Complexity, 2022, 35(5): 2004-2015.

Every univariate Hermite interpolation problem can be written as a pointwise limit of Lagrange interpolants.However,this property is not preserved for the multivariate case.In this paper,the authors first generalize the result of P.Gniadek.As an application,the authors consider the discrete approximation problem for a special case when the interpolation condition contains all partial derivatives of order less than n and one nth order differential polynomial.In addition,for the case of n ≥ 3,the authors use the concept of Cartesian tensors to give a sufficient condition to find a sequence of discrete points,such that the Lagrange interpolation problems at these points converge to the given Hermite-type interpolant.
[1] de Boor C and Ron A, On multivariate polynomial interpolation, Constr. Approx., 1990, 6(3):287-302.
[2] Gasca M and Sauer T, Polynomial interpolation in several variables, Adv. Comput. Math., 2000, 12(4):377-410.
[3] Gasca M and Sauer T, On the history of multivariate polynomial interpolation, Numerical Analysis Historical Developments in Century, 2001, 122(1):23-35.
[4] Lorentz R A, Multivariate Hermite interpolation by algebraic polynomials:A survey, J. Comput. Appl. Math., 2000, 122(1-2):167-201.
[5] Sauer T, Polynomial interpolation in several variables:Lattices, differences, and ideals, Studies in Computational Mathematics, 2006, 12:191-230.
[6] de Boor C, Ideal interpolation, Approximation Theory XI:Gatlinburg, Eds. by Chui C K, Neamtu M, and Schumaker L L, Nashboro Press, Brentwood, 2005, 59-91.
[7] Shekhtman B, Ideal interpolation:Translations to and from algebraic geometry, Approximate Commutative Algebra, Eds. by Robbiano L and Abbott J, Springer Vienna, 2010, 163-192.
[8] Chung K C and Yao T H, On lattices admitting unique Lagrange interpolations, SIAM J. Numer. Anal., 1977, 14:735-743
[9] Carnicer J and García-Esnaola M, Lagrange interpolation on conics and cubics, Comput. Aided Geom. Des., 2002, 19:313-326.
[10] Li Z, Zhang S, Dong T, et al., Error formulas for Lagrange projectors determined by Cartesian sets, Journal of Systems Science&Complexity, 2018, 31(4):1090-1102.
[11] Jiang X and Zhang S, The structure of the sencond-degree D-invariant subspace and its application in ideal interpolation, J. Approx. Theory, 2016, 207:232-240.
[12] Shekhtman B, On the limits of Lagrange projectors, Constr. Approx., 2009, 29(3):293-301.
[13] de Boor C, What are the limits of Lagrange projectors?, Constructive Theory of Functions, Varna, 2005, 51-63.
[14] de Boor C and Shekhtman B, On the pointwise limits of bivariate Lagrange projectors, Linear Algebra Appl., 2008, 429(1):311-325.
[15] Jiang X, Zhang S, and Shang B, The discretization for bivariate ideal interpolation, J. Comput. Appl. Math., 2016, 308:177-186.
[16] Gniadek P, Hermite multivariate interpolation as a limit of Lagrange interpolation, Univ. Iagell. Acra Math., 1999, 1236:127-138.
[17] Shekhtman B, On a conjecture of Carl de Boor regarding the limits of Lagrange interpolants, Constr. Approx., 2006, 24(3):365-370.
[18] Jiang X and Zhang S, The equivalent representation of the breadth-one D-invariant polynomial subspace and its discretization, Journal of Systems Science&Complexity, 2016, 29(5):1436-1445.
[19] Calvi J P, Intertwining unisolvent arrays for multivariate Lagrange interpolation, Adv. Comput. Math., 2005, 23(4):393-414.
[20] Comon P, Golub G, Lim L H, et al., Symmetric tensors and symmetric tensor rank, SIAM J. Matrix Anal. Appl., 2008, 30:1254-1279.
[21] Zhang R, A Brief Tutorial on Tensor Analysis, Tongji University Press, Shanghai, 2010(in Chinese).
[1] JIANG Xue,ZHANG Shugong,LI Zhe. An Algorithm for the Discretization of an Ideal Projector [J]. Journal of Systems Science and Complexity, 2016, 29(5): 1400-1410.
[2] JIANG Xue,ZHANG Shugong. The Equivalent Representation of the Breadth-One D-Invariant Polynomial Subspace and Its Discretization [J]. Journal of Systems Science and Complexity, 2016, 29(5): 1436-1445.
[3] MIN Guohua. ON SIMULTANEOUS L~P-APPROXIMATION (0 < p ≤ +∞) OF QUASI-HERMITE INTERPOLATION AND ITS DERIVATIVE [J]. Journal of Systems Science and Complexity, 1993, 6(3): 273-283.
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