Convergence of cascade algorithm for individual initial function and arbitrary refinement masks

The cascade algorithm plays an important role in computer graphics and wavelet analysis. For any initial function σ0, a cascade sequence (σn)∞n=1 is constructed by the iteration σn = Caσn-1,n = 1, 2, …, where Ca is defined by . 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(ϱn) for some ϱ ∈ (0,1). The condition of sum rules for the mask is not required. Finally, an example is presented to illustrate our theory.