Nonstationary Matrix Cascade Algorithms

This paper gives results on weak and strong convergence in L 2 (IR s) r of the cascade sequence (k;n) generated by the nonstation-ary matrix cascade algorithm k;n = jMj P j h k+1 (j) k+1;n?1 (M ?j); where for each k = 1; 2; : : : ; h k is a nite sequence of r r matrices and M is an integer dilation matrix. The limit as n ! 1 of the cascade sequence is an r-vector of functions (k) that is a solution of the nonsta-tionary matrix reenement equations k = jMj P j h k+1 (j) k+1 (M ?j): We give a characterization of weak convergence of (k;n) as n ! 1 under a weak assumption. Further assumption that there is a stationary reene-ment equation at 1 with matrix lter h satisfying P k jh k (j)?h(j)j < 1; for all j; gives convergence of the nonstationary cascade sequence (k;n) as both k and n tend to innnity. The convergence is completely determined by the spectral properties of the transition operator associated with h: x1. Introduction In 8] the authors considered weak and strong convergence in L 2 (IR s), s 1, of the cascade algorithm associated with a system of nonstationary scalar reenement equations and also derived results on the convergence of derivatives of the cascade sequence. In this paper we shall, in particular, extend some of these results to nonstationary matrix reenement equations. To write precisely these reenement equations we introduce some notation.