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

Abstract

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 $$C_a g = \sum\limits_{\alpha \in \mathbb{Z}} {a(\alpha )g(2 \cdot - \alpha ), g \in L_p (\mathbb{R}).} $$ . 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.