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Differentiability of multivariate refinable functions and factorization

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Abstract

The paper develops a necessary condition for the regularity of a multivariate refinable function in terms of a factorization property of the associated subdivision mask. The extension to arbitrary isotropic dilation matrices necessitates the introduction of the concepts of restricted and renormalized convergence of a subdivision scheme as well as the notion of subconvergence, i.e., the convergence of only a subsequence of the iterations of the subdivision scheme. Since, in addition, factorization methods pass even from scalar to matrix valued refinable functions, those results have to be formulated in terms of matrix refinable functions or vector subdivision schemes, respectively, in order to be suitable for iterated application. Moreover, it is shown for a particular case that the the condition is not only a necessary but also a sufficient one.

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Authors and Affiliations

  1. Lehrstuhl für Numerische Mathematik, Justus-Liebig-Universität Giessen, Heinrich-Buff-Ring 44, D-35392, Giessen, Germany

    Thomas Sauer

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  1. Thomas Sauer
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Communicated by T.N.T. Goodman

Dedicated to Charles A. Micchelli, a unique person, friend, mathematician and collaborator, on the occasion of his sixtieth birthday

Mathematics subject classifications (2000)

65T60, 65D99.

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Sauer, T. Differentiability of multivariate refinable functions and factorization. Adv Comput Math 25, 211–235 (2006). https://doi.org/10.1007/s10444-004-7635-y

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  • DOI: https://doi.org/10.1007/s10444-004-7635-y

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