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Subdivision Schemes and Norms

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Book cover Mathematical Methods for Curves and Surfaces (MMCS 2008)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5862))

Abstract

We compare three criteria of convergence of subdivision sche-mes for curves. The first involves the contraction of the differences of the refinements, the second, the contraction of the modulus of continuity of the refinements. The third one resorts to the notion of joint norm and may be used only for affine periodic subdivision schemes. We will relate these three criteria together by using specific norms for subdivision operators which make them equivalent for their respective scope. The duality theory plays an important role.

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References

  1. Buhmann, M.D., Micchelli, C.: Using two-slanted matrices for subdivision. Proc. London Math. Soc. 69(3), 428–448 (1994)

    Google Scholar 

  2. Charina, M., Conzi, C., Sauer, T.: L p -convergence of subdivision schemes: joint spectral radius versus restricted spectral radius. In: Chui, C.K., Neamtu, M., Schumaker, L.L. (eds.) Approximation Theory, Gatlinburg, TN, 2004, pp. 129–150. Nashboro Press, Brentwood (2005)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  4. Daubechies, I., Lagarias, J.C.: Two-scale difference equations II. Local regularity, infinite products of matrices and fractals. SIAM J. Math. Anal. 22, 1031–1079 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  5. Dubuc, S.: Scalar and Hermite subdivision schemes. Appl. Comp. Harm. Anal. 21, 376–394 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  6. Dyn, N., Gregory, J.A., Levin, D.: Analysis of linear binary subdivision schemes for curve design. Constr. Approx. 7, 127–147 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  7. Jia, R.-Q., Jiang, Q.: Spectral analysis of the transition operator and its application to smoothness anlysis of wavelets. SIAM J. Matrix Anal. Appl. 24, 1071–1109 (2003)

    Article  MATH  Google Scholar 

  8. Kantorovic, L.V., Akilov, G.P.: Functional Analysis in Normed Spaces. Pergamon, Oxford (1964)

    Google Scholar 

  9. Rota, G.-C., Strang, W.G.: A note on the joint spectral radius. Indag. Math. 22, 379–381 (1960)

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Département de mathématiques et de statistique, Université de Montréal, C.P. 6128, Succ. Centre-ville, Montréal, (Québec), Canada, H3C 3J7

    Serge Dubuc

Authors
  1. Serge Dubuc
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Editor information

Editors and Affiliations

  1. Department of Informatics, University of Oslo, Norway

    Morten Dæhlen  & Tom Lyche  & 

  2. Department of Informatics and Centre of Mathematics for Applications, University of Oslo, Norway

    Michael Floater

  3. INSA de Rennes, Rennes, France

    Jean-Louis Merrien

  4. Department of Informatics and Centre of Mathematics for Applications, University of Oslo, Oslo, Norway

    Knut Mørken

  5. Department of Mathematics, Vanderbilt University, 37240, Nashville, TN, USA

    Larry L. Schumaker

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Dubuc, S. (2010). Subdivision Schemes and Norms. In: Dæhlen, M., Floater, M., Lyche, T., Merrien, JL., Mørken, K., Schumaker, L.L. (eds) Mathematical Methods for Curves and Surfaces. MMCS 2008. Lecture Notes in Computer Science, vol 5862. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11620-9_12

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  • DOI: https://doi.org/10.1007/978-3-642-11620-9_12

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-11619-3

  • Online ISBN: 978-3-642-11620-9

  • eBook Packages: Computer ScienceComputer Science (R0)

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