The cascade algorithm plays an important role in computer graphics and wavelet analysis. For any initial function φ0, a cascade sequence (φn)∞n=1 is constructed by the iteration φn=Caφn-1,n=1,2,…,where Ca is defined by Cag=∑a(α)g(2·-α)，g∈Lp(R).a∈Z In this paper, we characterize the convergence of a cascade sequence in terms of a sequence of functions and in terms of joint spectral radius. As a consequence, it is proved that any convergent cascade sequence has a convergence rate of geometry, i.e.,||φn+1 - φn||Lp(R)= O(n) for some ∈ (0, 1). The condition of sum rules for the mask is not required. Finally, an example is presented to illustrate our theory.
Starting with an initial vector λ = (λ(κ))κ∈z ∈ ep(Z), the subdivision scheme generates asequence (Snaλ)∞n=1 of vectors by the subdivision operator Saλ(κ) = ∑λ(j)a(k - 2j), k ∈ Z. j∈zSubdivision schemes play an important role in computer graphics and wavelet analysis. It is very interesting tounderstand under what conditions the sequence (Snaλ)∞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 forany initial vector in ep(Z) provided that it does for one nonzero vector in that space. Moreover, if the integertranslates of the refinable function are stable, the smoothness of the limit function corresponding to the vectorλ is also independent of λ.
In this paper, a large class of n dimensional orthogonal and biorthognal wavelet filters (lowpass and highpass) are presented in explicit expression. We also characterize orthogonal filters with linear phase in this case. Some examples are also given, including non separable orhogonal and biorthogonal filters with linear phase.
Si-long Peng (NADEC, Institute of Automation, Chinese Academy of Sciences, Beijing 100080, China)
