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Dual Hermite subdivision schemes of de Rham-type

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Abstract

Though a Hermite subdivision scheme is non-stationary by nature, its non-stationarity can be of two types, making useful the distinction between Inherently Stationary and Inherently Non-Stationary Hermite subdivision schemes. This paper focuses on the class of inherently stationary, dual non-interpolatory Hermite subdivision schemes that can be obtained from known Hermite interpolatory ones, by applying a generalization of the de Rham corner cutting strategy. Exploiting specific tools for the analysis of inherently stationary Hermite subdivision schemes we show that, giving up the interpolation condition, the smoothness of the associated basic limit function can be increased by one, while its support width is only enlarged by one. To accomplish the analysis of de Rham-type Hermite subdivision schemes two new theoretical results are derived and the new notion of \(HC^{\ell }\)-convergence is introduced. It allows the construction of Hermite-type subdivision schemes of order \(d+1\) with the first element of the vector valued limit function having regularity \(\ell \ge d\).

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References

  1. Charina, M., Conti, C., Sauer, T.: Regularity of multivariate vector subdivision schemes. Numer. Algorithms 39, 97–113 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  2. Cohen, A., Dyn, N., Levin, D.: Stability and inter-dependence of matrix subdivision schemes. In: Fontanella, F., Jetter, K., Laurent, P.-J. (eds.) Advanced Topics in Multivariate Approximation, pp. 33–46. World Scientific Publishing Co. Inc., Singapore (1993)

    Google Scholar 

  3. Conti, C., Hormann, K.: Polynomial reproduction for univariate subdivision schemes of any arity. J. Approx. Theory 163, 413–437 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  4. Conti, C., Jetter, K.: A new subdivision method for bivariate splines on the four-directional mesh. J. Comput. Appl. Math. 119, 936–952 (2000)

    MathSciNet  Google Scholar 

  5. Conti, C., Romani, L.: Algebraic conditions on non-stationary subdivision symbols for exponential polynomial reproduction. J. Comput. Appl. Math. 236, 543–556 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  6. Conti, C., Romani, L.: Dual univariate \(m\)-ary subdivision schemes of de Rham-type. J. Math. Anal. Appl. 407, 443–456 (2013)

    Article  MathSciNet  Google Scholar 

  7. Conti, C., Zimmermann, G.: Interpolatory rank-1 vector subdivision schemes. Comput. Aided Geom. Des. 21, 341–351 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  8. de Rham, G.: Sur une courbe plane. J. Math. Pures Appl. 35, 25–42 (1956)

    MATH  MathSciNet  Google Scholar 

  9. Dubuc, S., Merrien, J.-L., de Rham, G.:Transform of a Hermite subdivision scheme. In: Neamtu, M., Schumaker, L.L. (eds.) Approximation Theory XII, San Antonio 2007, pp. 121–132. Nashboro Press, Nashville (2008)

  10. Dubuc, S., Merrien, J.L.: Hermite subdivision schemes and Taylor polynomials. Constr. Approx. 29, 219–245 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  11. Dyn, N., Levin, D.: Analysis of Hermite-interpolatory subdivision schemes, In: Dubuc, S., Deslauriers, G. (eds.) Spline Functions and the Theory of Wavelets, pp. 105–113. American Mathematical Society, Providence (1999)

  12. Guglielmi, N., Manni, C., Vitale, D.: Convergence analysis of \(C^2\) Hermite interpolatory subdivision schemes by explicit joint spectral radius formulas. Linear Algebra Appl. 434, 884–902 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  13. Han, B.: Vector cascade algorithms and refinable functions in Sobolev spaces. J. Approx. Theory 124, 44–88 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  14. Han, B., Yu, T.P., Piper, B.: Multivariate refinable Hermite interpolant. Math. Comput. 73(248), 1913–1935 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  15. Han, B., Yu, T.P., Xue, Y.: Noninterpolatory Hermite subdivision schemes. Math. Comput. 74(251), 1345–1367 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  16. Merrien, J.-L.: A family of Hermite interpolants by bisection algorithms. Numer. Algorithms 2, 187–200 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  17. Merrien, J.-L.: Interpolants d’Hermite \(C^2\) obtenus par subdivision. M2AN Math. Model. Numer. Anal. 33, 55–65 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  18. Merrien, J.-L., Sauer, T.: A generalized Taylor factorization for Hermite subdivision schemes. J. Comput. Appl. Math. 236(4), 565–574 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  19. Micchelli, C.A., Sauer, T.: On vector subdivision. Math. Z. 229, 621–674 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  20. Romani, L.: A circle-preserving \(C^2\) Hermite interpolatory subdivision scheme with tension control. Comput. Aided Geom. Des. 27, 36–47 (2010)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgments

This work was partially supported by Italian funds from INdAM-GNCS. Special thanks go to the anonymous referees for their useful comments.

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Authors and Affiliations

  1. Dipartimento di Ingegneria Industriale, Università di Firenze, Viale Morgagni 40/44, 50134 , Florence, Italy

    Costanza Conti

  2. INSA de Rennes, 20 Avenue des Buttes de Coesmes, CS 14315, 35043 , Rennes, France

    Jean-Louis Merrien

  3. Dipartimento di Matematica e Applicazioni, Università di Milano-Bicocca, Via R. Cozzi 55, 20125 , Milan, Italy

    Lucia Romani

Authors
  1. Costanza Conti
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  2. Jean-Louis Merrien
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  3. Lucia Romani
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Correspondence to Lucia Romani.

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Communicated by Michael Vogelius.

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Conti, C., Merrien, JL. & Romani, L. Dual Hermite subdivision schemes of de Rham-type. Bit Numer Math 54, 955–977 (2014). https://doi.org/10.1007/s10543-014-0495-z

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