Convergence of subdivision schemes and smoothness of limit functions

Abstract

Starting with 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 S a λ ( k ) = ∑ j ∈ Z λ ( j ) a ( k − 2 j ) , k ∈ 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 a n λ } n = 1 ∞ converges to a L p -function in an appropriate sense. This problem has been studied extensively. In this paper, we consider the convergence of subdivision scheme in Sobolev spaces with the tool of joint spectral radius. Firstly, the conditions under which the sequence { S a n λ } n = 1 ∞ converges to a W p k -function in an appropriate sense are given. Then, we show that the subdivision scheme converges for any initial vector in W p k ( R ) provided that it does for one nonzero vector in that space. Moreover, if the shifts of the refinable function are stable, the smoothness of the limit function corresponding to the vector λ is also independent of λ, where the smoothness of a given function is measured by the generalized Lipschitz space.