CASCADE ALGORITHMS IN WAVELET ANALYSIS
Rong-Qing Jia · 2002
In this paper we survey some recent results on cascade algorithms. Let a be a finitely supported sequence on Z. The cascade operator Qa is the the linear operator on Lp(IR) (1 ≤ p ≤ ∞) given by Qaf: = � a(j)f(2 · − j), f ∈ Lp(IR). j ∈ Z The iteration scheme Q n af (n = 1, 2,...) is called the cascade algorithm associated with a. The Lp convergence of a cascade algorithm is characterized in terms of the p-norm joint spectral radius of two matrices associated with the corresponding mask. For the special case p = 2, convergence of a cascade algorithm is characterized in terms of the spectrum of the transition matrix associated with the mask. Then the basic theory on cascade algorithms is employed to give a unified treatment of orthogonal wavelets, biorthogonal wavelets, and fundamental refinable functions. Furthermore, we give a comprehensive review of biorthogonal wavelet bases. Our methods can be used to deal with more complicated