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On polynomial symbols for subdivision schemes

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Abstract

Given a dilation matrix A :ℤd→ℤd, and G a complete set of coset representatives of 2π(A −Tℤd/ℤd), we consider polynomial solutions M to the equation ∑ g∈G M(ξ+g)=1 with the constraints that M≥0 and M(0)=1. We prove that the full class of such functions can be generated using polynomial convolution kernels. Trigonometric polynomials of this type play an important role as symbols for interpolatory subdivision schemes. For isotropic dilation matrices, we use the method introduced to construct symbols for interpolatory subdivision schemes satisfying Strang–Fix conditions of arbitrary order.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Aalborg University, Fredrik Bajers Vej 7G, DK-9220, Aalborg East, Denmark

    Morten Nielsen

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  1. Morten Nielsen
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Correspondence to Morten Nielsen.

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Communicated by T. Sauer

Research partially supported by the Danish Technical Science Foundation, Grant No. 9701481, and by the Danish SNF-PDE network.

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Nielsen, M. On polynomial symbols for subdivision schemes. Adv Comput Math 27, 195–209 (2007). https://doi.org/10.1007/s10444-005-7476-3

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  • DOI: https://doi.org/10.1007/s10444-005-7476-3

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