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 phi(0), a cascade sequence (phi(n)) (infinity)(n=1) is constructed by the iteration phi(n) = C-a phi(n-1,)n = 1, 2,... where Ca is defined by C(a)g = Sigma (alpha is an element of Z) a(alpha)g(2(.)-alpha), g is an element of L-p(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., ||phi(n+1)-phi(n)||(Lp(R)) = O (rho(n)) for some rho is an element of (0,1). The condition of sum rules for the mask is not required. Finally, an example is presented to illustrate our theory.