Convergence of nonstationary cascade algorithms

A nonstationary multiresolution of L 2 (R s) is generated by a sequence of scaling functions k 2 L 2 (R s); k 2 Z: We consider (k) that is the solution of the nonstationary reenement equations k = jMj P j h k+1 (j) k+1 (M ?j); k 2 Z; where h k is nitely supported for each k and M is a dilation matrix. We study various forms of convergence in L 2 (R s) of the corresponding nonstationary cascade algorithm k;n = jMj P j h k+1 (j) k+1;n?1 (M ?j); as k or n tends to 1: It is assumed that there is a stationary reenement equation at 1 with lter sequence h and that P k jh k (j) ? h(j)j < 1: The results show that the convergence of the nonstationary cascade algorithm is determined by the spectral properties of the transition operator associated with h: