Convergence of nonstationary cascade algorithms

A nonstationary multiresolution of L-2(R-s) is generated by a sequence of s caling 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 equ ations 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 nonst ationary 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 stationar y 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 nonstat ionary cascade algorithm is determined by the spectral properties of the tr ansition operator associated with h. Mathematics Subject Classification (19 91): 41A15, 41A30, 42C05, 42C15.