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Polynomial interpolation, ideals and approximation order of multivariate refinable functions

Author: Thomas Sauer
Journal: Proc. Amer. Math. Soc. 130 (2002), 3335-3347
MSC (2000): Primary 42C40, 13P10; Secondary 41A05
Published electronically: May 29, 2002
MathSciNet review: 1913013
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Abstract: The paper identifies the multivariate analog of factorization properties of univariate masks for compactly supported refinable functions, that is, the ``zero at $\pi$''-property, as containment of the mask polynomial in an appropriate quotient ideal. In addition, some of these quotient ideals are given explicitly.

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  • 1. G. Birkhoff.
    The algebra of multivariate interpolation.
    In C.V. Coffman and G.J. Fix, editors, Constructive Approaches to Mathematical Models, pages 345-363. Academic Press, Inc., 1979. MR 83d:41001
  • 2. C. de Boor and A. Ron.
    On multivariate polynomial interpolation.
    Constr. Approx., 6 (1990), 287-302. MR 91c:41005
  • 3. C. de Boor and A. Ron.
    The least solution for the polynomial interpolation problem.
    Math. Z., 210 (1992), 347-378. MR 93f:41002
  • 4. A. S. Cavaretta, W. Dahmen, and C. A. Micchelli.
    Stationary Subdivision, volume 93 (453) of Memoirs of the AMS.
    Amer. Math. Soc., 1991. MR 92h:65017
  • 5. D. Cox, J. Little, and D. O'Shea.
    Ideals, Varieties and Algorithms.
    Undergraduate Texts in Mathematics. Springer-Verlag, 2nd edition, 1996. MR 97h:13024
  • 6. W. Dahmen and C. A. Micchelli.
    Local dimension of piecewise polynomial spaces, syzygies, and the solutions of systems of partial differential operators.
    Math. Nachr., 148 (1990), 117-136. MR 92i:35029
  • 7. I. Daubechies.
    Ten Lectures on Wavelets, volume 61 of CBMS-NSF Regional Conference Series in Applied Mathematics.
    SIAM, 1992. MR 93e:42045
  • 8. B. Han and R.-Q. Jia.
    Multivariate refinement equations and convergence of subdivision schemes.
    SIAM J. Math. Anal., (1998), 1177-1199. MR 99f:41018
  • 9. K. Jetter and G. Plonka.
    A survey on $L_2$-approximation order from shift-invariant spaces.
    Technical Report SM-DU-441, Gerhard Mercator Universität Gesamthochschule Duisburg, 1999.
  • 10. V. Latour, J. Müller, and W. Nickel.
    Stationary subdivision for general scaling matrices.
    Math. Z., 227 (1998), 645-661. MR 99c:65027
  • 11. H. M. Möller.
    Hermite interpolation in several variables using ideal-theoretic methods.
    In W. Schempp and K. Zeller, editors, Constructive Theory of Functions of Several Variables, Lecture Notes in Mathematics, pages 155-163, Springer, 1977. MR 58:12087
  • 12. H. M. Möller and T. Sauer.
    H-bases for polynomial interpolation and system solving.
    Advances Comput. Math., 12 (2000), 335-362. MR 2001g:41005
  • 13. H. M. Möller and T. Sauer.
    H-bases I: The foundation.
    In A. Cohen, C. Rabut, and L. L. Schumaker, editors, Curve and Surface fitting: Saint-Malo 1999, pages 325-332, Vanderbilt University Press, 2000.
  • 14. B. Renschuch.
    Elementare und praktische Idealtheorie.
    VEB Deutscher Verlag der Wissenschften, 1976. MR 56:2981
  • 15. T. Sauer.
    Polynomial interpolation of minimal degree and Gröbner bases.
    In B. Buchberger and F. Winkler, editors, Groebner Bases and Applications (Proc. of the Conf. 33 Years of Groebner Bases), volume 251 of London Math. Soc. Lecture Notes, pages 483-494, Cambridge University Press, 1998. MR 2000h:13019
  • 16. T. Sauer.
    Gröbner bases, H-bases and interpolation.
    Trans. Amer. Math. Soc., 353 (2001), 2293-2308. MR 2002b:13035

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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: Subdivision, polynomial preservation, refinable functions, quotient ideals
Received by editor(s): October 26, 2000
Received by editor(s) in revised form: June 19, 2001
Published electronically: May 29, 2002
Additional Notes: This work was supported by Deutsche Forschungsgemeinschaft with a Heisenberg fellowship, Grant # Sa 627/6–1
Dedicated: Dedicated to C. A. Micchelli on the occasion of his 60th birthday, with friendship and gratitude for a wonderful collaboration.
Communicated by: David R. Larson
Article copyright: © Copyright 2002 American Mathematical Society

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