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From Hermite to stationary subdivision schemes in one and several variables

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Abstract

Vector and Hermite subdivision schemes both act on vector data, but since the latter one interprets the vectors as function values and consecutive derivatives they differ by the “renormalization” of the Hermite scheme in any step. In this paper we give an algebraic factorization method in one and several variables to relate any Hermite subdivision scheme that satisfies the so–called spectral condition to a vector subdivision scheme. These factorizations are natural extensions of the “zero at π” condition known for the masks of refinable functions. Moreover, we show how this factorization can be used to investigate different forms of convergence of the Hermite scheme and why the multivariate situation is conceptionally more intricate than the univariate one. Finally, we give some examples of such factorizations.

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Further Reading

  1. Dubuc, S., Merrien, J.-L.: Convergent vector and Hermite subdivision schemes. Constr. Approx. 23, 1–22 (2006)

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Authors and Affiliations

  1. INSA de Rennes, 20 av. des Buttes de Coesmes, CS 14315, 35043, Rennes Cedex, France

    Jean–Louis Merrien

  2. Lehrstuhl für Numerische Mathematik, Justus–Liebig–Universität Gießen, Heinrich-Buff-Ring 44, 35392, Gießen, Germany

    Tomas Sauer

Authors
  1. Jean–Louis Merrien
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  2. Tomas Sauer
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Correspondence to Tomas Sauer.

Additional information

Communicated by T. N. T. Goodman.

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Merrien, J., Sauer, T. From Hermite to stationary subdivision schemes in one and several variables. Adv Comput Math 36, 547–579 (2012). https://doi.org/10.1007/s10444-011-9190-7

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  • DOI: https://doi.org/10.1007/s10444-011-9190-7

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