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On a conjecture of Tomas Sauer regarding nested ideal interpolation

Author: Boris Shekhtman
Journal: Proc. Amer. Math. Soc. 137 (2009), 1723-1728
MSC (2000): Primary 41A63; Secondary 41A10, 41A80, 13P10
Published electronically: December 11, 2008
MathSciNet review: 2470830
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Abstract: Tomas Sauer conjectured that if an ideal complements polynomials of degree less than $ n$, then it is contained in a larger ideal that complements polynomials of degree less than $ n-1$. We construct a counterexample to this conjecture for polynomials in three variables and with $ n=3$.

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  • 1. G. Birkhoff, The algebra of multivariate interpolation, Constructive approaches to mathematical models (Proc. Conf. in honor of R. J. Duffin, Pittsburgh, Pa., 1978) (C. V. Coffman and G. J. Fix, eds.), Academic Press, New York, NY, 1979, pp. 345-363. MR 559505 (83d:41001)
  • 2. C. de Boor, Ideal interpolation, Approximation Theory XI: Gatlinburg, 2004 (C. K. Chui, M. Neamtu, and L. Schumaker, eds.), Mod. Methods Math., Nashboro Press, Brentwood, TN, 2005, pp. 59-91. MR 2126674 (2005k:41004)
  • 3. C. de Boor and A. Ron, On polynomial ideals of finite codimension with applications to box spline theory, J. Math. Anal. Appl. 158 (1991), no. 1, 168-193. MR 1113408 (93a:41014)
  • 4. C. de Boor and B. Shekhtman, On the pointwise limits of bivariate Lagrange projectors, Linear Algebra Appl. 429 (2008), no. 1, 311-325. MR 2419159
  • 5. F. S. Macaulay, The algebraic theory of modular systems, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1994, revised reprint of the 1916 original, with an introduction by Paul Roberts. MR 1281612 (95i:13001)
  • 6. Hans Michael Möller, Hermite interpolation in several variables using ideal-theoretic methods, Constructive theory of functions of several variables (Proc. Conf., Math. Res. Inst., Oberwolfach, 1976), Lecture Notes in Math., vol. 571, Springer, Berlin, 1977, pp. 155-163. MR 0493046 (58:12087)
  • 7. K. C. O'Meara and C. Vinsonhaler, On approximately simultaneously diagonalizable matrices, Linear Algebra Appl. 412 (2006), no. 1, 39-74. MR 2180858 (2006g:15022)
  • 8. T. Sauer, Polynomial interpolation in several variables: Lattices, differences, and ideals, Topics in Multivariate Approximation and Interpolation (M. Buhmann, W. Hausmann, K. Jetter, R. Schaback, and J. Stöckler, eds.), Studies in Computational Mathematics, vol. 12, Elsevier, Amsterdam, The Netherlands, 2005, pp. 189-228.
  • 9. T. Sauer and Yuan Xu, On multivariate Hermite interpolation, Advances Comput. Math. 4 (1995), no. 4, 207-259. MR 1357718 (96j:41031)
  • 10. -, On multivariate Lagrange interpolation, Math. Comp. 64 (1995), 1147-1170. MR 1297477 (95j:41051)

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Additional Information

Boris Shekhtman
Affiliation: Department of Mathematics and Statistics, University of South Florida, Tampa, Florida 33620

Keywords: Ideal interpolation, nested ideals, multivariate divided differences
Received by editor(s): May 30, 2008
Published electronically: December 11, 2008
Communicated by: Nigel J. Kalton
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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