Convergence of nonstationary cascade algorithms

Abstract

A nonstationary multiresolution of L2(ℝS) is generated by a sequence of scaling functions φk ∈ L2(ℝS), k ∈ ℤ. We consider (φk) that is the solution of the nonstationary refinement equations φk = |M| ∑J hk+1(j)φk+1(M · -j), k ∈ ℤ, where hk is finitely supported for each k and M is a dilation matrix. We study various forms of convergence in L2(ℝS) of the corresponding nonstationary cascade algorithm φk,n = |M| ∑J hk+1(j)φk+1,n-1 (M · -j), as k or n tends to ∫. It is assumed that there is a stationary refinement equation at ∫ with filter sequence h and that ∑|hk(j) - h(j)| < ∫. The results show that the convergence of the nonstationary cascade algorithm is determined by the spectral properties of the transition operator associated with h. © Springer-Verlag 1999.