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Lagrange interpolation on subgrids of tensor product grids

Author: Thomas Sauer
Journal: Math. Comp. 73 (2004), 181-190
MSC (2000): Primary 54C40, 14E20; Secondary 46E25, 20C20
Published electronically: June 6, 2003
MathSciNet review: 2034116
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Abstract: This note shows that a wide class of algebraically motivated constructions for Lagrange interpolation polynomials always yields a tensor product interpolation space as long as the nodes form a tensor product grid or a lower subset thereof.

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  • 1. J. Bauschinger, Interpolation, Encyklopädie der Mathematischen Wissenschaften, Band I, Teil 2, B. G. Teubner, Leipzig, 1900, pp. 800-821.
  • 2. O. Biermann, Über näherungsweise Kubaturen, Monatsh. Math. Phys. 14 (1903), 211-225.
  • 3. C. de Boor and A. Ron, On multivariate polynomial interpolation, Constr. Approx. 6 (1990), 287-302. MR 91c:41005
  • 4. -, Computational aspects of polynomial interpolation in several variables, Math. Comp. 58 (1992), no. 198, 705-727. MR 92i:65022
  • 5. -, The least solution for the polynomial interpolation problem, Math. Z. 210 (1992), no. 210, 347-378. MR 93f:41002
  • 6. B. Buchberger, Ein algorithmus zum auffinden der basiselemente des restklassenrings nach einem nulldimensionalen polynomideal, Ph.D. thesis, Innsbruck, 1965.
  • 7. -, An algorithmic criterion for the solvability of algebraic systems of equations (german), Aequationes Math. 4 (1970), no. 3, 374-383. MR 42:3077
  • 8. B. Buchberger and H. M. Möller, The construction of multivariate polynomials with preassigned zeros, Computer Algebra, EUROCAM '82, European Computer Algebra Conference (G. Goos and J. Hartmanis, eds.), Lecture Notes in Computer Science, vol. 144, Springer Verlag, 1982, pp. 24-31. MR 84b:12003
  • 9. D. Cox, J. Little, and D. O'Shea, Ideals, varieties and algorithms, 2. ed., Undergraduate Texts in Mathematics, Springer-Verlag, 1996. MR 97h:13024
  • 10. D. Eisenbud, Commutative algebra with a view toward algebraic geometry, Graduate Texts in Mathematics, vol. 150, Springer, 1994. MR 97a:13001
  • 11. M. Gasca and T. Sauer, On the history of multivariate polynomial interpolation, J. Comput. Appl. Math. 122 (2000), 23-35.
  • 12. -, Polynomial interpolation in several variables, Advances Comput. Math. 12 (2000), 377-410. MR 2001d:41010
  • 13. W. Gröbner, Algebraische geometrie II, B.I-Hochschultaschenbücher, no. 737, Bibliographisches Institut Mannheim, 1970. MR 48:8499
  • 14. L. Kronecker, Über einige Interpolationsformeln für ganze Funktionen mehrerer Variabeln, L. Kroneckers Werke (H. Hensel, ed.), Teubner / Chelsea Publishing Company, 1895 / 1966, Lecture at the Academy of Sciences, December 21, 1865, pp. 133-141. MR 38:5576
  • 15. F. S. Macaulay, The algebraic theory of modular systems, Cambridge Tracts in Math. and Math. Physics, no. 19, Cambridge Univ. Press, 1916. MR 95k:13001 (reprint)
  • 16. H. M. Möller and T. Sauer, H-bases for polynomial interpolation and system solving, Advances Comput. Math. 12 (2000), 335-362. MR 2001g:41005
  • 17. -, H-bases I: The foundation, Curve and Surface Fitting: Saint-Malo 1999 (A. Cohen, C. Rabut, and L. L. Schumaker, eds.), Vanderbilt University Press, 2000, pp. 325-332.
  • 18. T. Sauer, Polynomial interpolation of minimal degree, Numer. Math. 78 (1997), no. 1, 59-85. MR 99f:41042
  • 19. -, Polynomial interpolation of minimal degree and Gröbner bases, Groebner Bases and Applications (Proc. of the Conf. 33 Years of Groebner Bases) (B. Buchberger and F. Winkler, eds.), London Math. Soc. Lecture Notes, vol. 251, Cambridge University Press, 1998, pp. 483-494. MR 2000h:13019
  • 20. -, Gröbner bases, H-bases and interpolation, Trans. Amer. Math. Soc. 353 (2001), 2293-2308. MR 2002b:13035
  • 21. H. Werner, Remarks on Newton type multivariate interpolation for subsets of grids, Computing 25 (1980), 181-191. MR 82e:65010

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Additional Information

Thomas Sauer
Affiliation: Lehrstuhl für Numerische Mathematik, Justus–Liebig–Universität Gießen, Heinrich–Buff–Ring 44, D–35392 Gießen, Germany

Keywords: Interpolation, Gr\"obner bases, H-bases, tensor products
Received by editor(s): January 31, 2001
Received by editor(s) in revised form: June 8, 2002
Published electronically: June 6, 2003
Dedicated: Dedicated to Mariano Gasca, without whom the field of polynomial interpolation would be very much depleted, on the occasion of his 60th birthday.
Article copyright: © Copyright 2003 American Mathematical Society

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