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Refinement equations and spline functions

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Abstract

In this paper, we exploit the relation between the regularity of refinable functions with non-integer dilations and the distribution of powers of a fixed number modulo 1, and show the nonexistence of a non-trivial C  ∞  solution of the refinement equation with non-integer dilations. Using this, we extend the results on the refinable splines with non-integer dilations and construct a counterexample to some conjecture concerning the refinable splines with non-integer dilations. Finally, we study the box splines satisfying the refinement equation with non-integer dilation and translations. Our study involves techniques from number theory and harmonic analysis.

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Authors and Affiliations

  1. Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, Vilnius, 03225, Lithuania

    Artūras Dubickas

  2. Institute of Mathematics and Informatics, Akademijos 4, Vilnius, 08663, Lithuania

    Artūras Dubickas

  3. Institute of Computational Mathematics and Science and Engineering Computing, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing, 100080, China

    Zhiqiang Xu

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  1. Artūras Dubickas
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  2. Zhiqiang Xu
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Correspondence to Zhiqiang Xu.

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Communicated by R. Q. Jia.

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Dubickas, A., Xu, Z. Refinement equations and spline functions. Adv Comput Math 32, 1 (2010). https://doi.org/10.1007/s10444-008-9079-2

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