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The independence of initial vectors in the subdivision schemes

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Abstract

Starting with an initial vector λ=(λ(k))k∈ℤ ∈ ℓp(ℤ), the subdivision scheme generates a sequence (S n a λ) ∞n=1 of vectors by the subdivision operator

$$S_a \lambda (k) = \sum\limits_{j \in \mathbb{Z}} {\lambda (j)a(k - 2j)} , k \in \mathbb{Z}$$

. Subdivision schemes play an important role in computer graphics and wavelet analysis. It is very interesting to understand under what conditions the sequence (S n a λ) ∞n=1 converges to an Lp-function in an appropriate sense. This problem has been studied extensively. In this paper we show that the subdivision scheme converges for any initial vector in ℓp(ℤ) provided that it does for one nonzero vector in that space. Moreover, if the integer translates of the refinable function are stable, the smoothness of the limit function corresponding to the vector A is also independent of λ.

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Authors and Affiliations

  1. Department of Applied Mathematics, Beijing University of Aeronautics and Astronautics, 100083, Beijing, China

    Chen Dirong

  2. Department of Mathematics, Hubei Institute for Nationalities, 445000, Enshi, China

    Chen Dirong

  3. Department of Mathematics, Hubei University, 430062, Wuhan, China

    Li Luoqing

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  1. Chen Dirong
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  2. Li Luoqing
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Correspondence to Chen Dirong.

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Dirong, C., Luoqing, L. The independence of initial vectors in the subdivision schemes. Sci. China Ser. A-Math. 45, 1439–1445 (2002). https://doi.org/10.1007/BF02880038

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  • DOI: https://doi.org/10.1007/BF02880038

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