Convergence of nonstationary cascade algorithms

Abstract

A nonstationary multiresolution of $L^2(\mathbb{R}^s)$ is generated by a sequence of scaling functions $\phi_k\in L^2(\mathbb{R}^s), k\in \mathbb{Z}.$ We consider $(\phi_k)$ that is the solution of the nonstationary refinement equations $\phi_k = |M|$ $ \sum_{j} h_{k+1}(j)\phi_{k+1}(M \cdot -j), k\in \mathbb{Z},$ where $h_k$ is finitely supported for each k and M is a dilation matrix. We study various forms of convergence in $L^2(\mathbb{R}^s)$ of the corresponding nonstationary cascade algorithm $\phi_{k,n} = |M| \sum_{j} h_{k+1}(j)\phi_{k+1,n-1}(M \cdot -j),$ as k or n tends to $\infty.$ It is assumed that there is a stationary refinement equation at $\infty$ with filter sequence h and that $\sum_k |h_k(j) - h(j)| < \infty.$ The results show that the convergence of the nonstationary cascade algorithm is determined by the spectral properties of the transition operator associated with h.