Convergence of nonstationary cascade algorithms

A nonstationary multiresolution of L-2(R-s) is generated by a sequence of scaling functions phi(k) is an element of L-2(R-s), k is an element of Z. We consider (phi(k)) that is the solution of the nonstationary refinement equations phi(k) = /M/ Sigma(j)(h)k+1 (j)phi(k+1) (M. -j), k is an element of Z; where h(k) is finitely supported for each k and M is a dilation matrix. We study various forms of convergence in L2(R-s) Of the corresponding nonstationary cascade algorithm phi(k,n) = /M/ Sigma(j)(h)k+1 (j)phi(k+1,n-1) (M. -j), as k or n tends to infinity. It is assumed that there is a stationary refinement equation at oo with filter sequence h and thar Sigma(k) /h(k)(j) - h(j)/ < infinity, The results show that the convergence of the nonstationary cascade algorithm is determined by the spectral properties of the transition operator associated with h. Mathematics Subject Classification (1991): 41A15, 41A30, 42C05, 42C15.