The independence of initial vectors in the subdivision schemes

Starting with an initial vector λ=(λ(k)) ^k∈ℤ ∈ ℓ_p(ℤ), the subdivision scheme generates a sequence (S _a ⁿ λ) _n=1 ^∞ of vectors by the subdivision operator. Subdivision schemes play an important role in computer graphics and wavelet analysis. It is very interesting to understand under what conditions the sequence (S _a ⁿ λ) _n=1 ^∞ converges to an L_p-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 ℓ_p(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.