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Commutator Estimate for Nonlinear Subdivision

  • Conference paper
Mathematical Methods for Curves and Surfaces (MMCS 2012)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8177))

Abstract

Nonlinear multiscale algorithms often involve nonlinear perturbations of linear coarse-to-fine prediction operators S (also called subdivision operators). In order to control these perturbations, estimates of the “commutator” SF − FS of S with a sufficiently smooth map F are needed. Such estimates in terms of bounds on higher-order differences of the underlying mesh sequences have already appeared in the literature, in particular in connection with manifold-valued multiscale schemes. In this paper we give a compact (and in our opinion technically less tedious) proof of commutator estimates in terms of local best approximation by polynomials instead of bounds on differences covering multivariate S with general dilation matrix M. An application to the analysis of normal multiscale algorithms for surface representation is outlined.

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Authors and Affiliations

  1. SES, Jacobs University, 28759, Bremen, Germany

    Peter Oswald & Tatiana Shingel

Authors
  1. Peter Oswald
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  2. Tatiana Shingel
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Editor information

Editors and Affiliations

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

    Michael Floater

  2. DMA, University of Oslo, P.O.Box 1053, 0316, Blindern, Oslo, Norway

    Tom Lyche

  3. Laboratoire LJK, Université Joseph Fourier, BP 53, 38041, Grenoble Cedex 9, France

    Marie-Laurence Mazure

  4. University of Oslo, Norway

    Knut Mørken

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

    Larry L. Schumaker

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Oswald, P., Shingel, T. (2014). Commutator Estimate for Nonlinear Subdivision. In: Floater, M., Lyche, T., Mazure, ML., Mørken, K., Schumaker, L.L. (eds) Mathematical Methods for Curves and Surfaces. MMCS 2012. Lecture Notes in Computer Science, vol 8177. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54382-1_22

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  • DOI: https://doi.org/10.1007/978-3-642-54382-1_22

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-54381-4

  • Online ISBN: 978-3-642-54382-1

  • eBook Packages: Computer ScienceComputer Science (R0)

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