Nonstationary Matrix Cascade Algorithms

摘要

This paper gives results on weak and strong convergence in L~2 (IR~s)~r of the cascade sequence (PHI_(k, n)) generated by the nonstation-ary matrix cascade algorithm PHI_(k, n) chemical bounds|M| sum_j h_k+1(j) PHI_(k+1,n-1) (M contre dot -j), where for each k chemical bounds 1,2,-, h_k is finite sequence of r x r matrices and M is an integer dilation matrix. The limit as n -> infinity of the cascade sequence is an r-vector of functions (PHI_k) that is a solution of the nonstationary matrix refinement equations PHI_k chemical bounds |M| sum_j h_(k+1)(j) PHI_(k+1)(M centre dot -j). We give a characterization of weak convergence of (PHI_(k, n)) as n -> infinity under a weak assumption. Further assumption that there is a stationary refinement equation at infinity with matrix filter h satisfying sum_k |h_k(j) - h(j) | < infinity, for all j, gives convergence of the nonstationary cascade sequence (PHI_(k,n)) as both k and n tend to infinity. The convergence is completely determined by the spectral properties of the transition operator associated with h.