Differentiability of multivariate refinable functions and factorization |
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Authors: | Thomas Sauer |
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Institution: | 001. Lehrstuhl für Numerische Mathematik, Justus-Liebig-Universit?t Giessen, Heinrich-Buff-Ring 44, D-35392, Giessen, Germany
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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.
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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Keywords: | subdivision refinable function subconvergence |
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