Summary.
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.
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Received September 19, 1997 / Revised version received May 22, 1998 / Published online August 19, 1999
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Goodman, T., Lee, S. Convergence of nonstationary cascade algorithms. Numer. Math. 84, 1–33 (1999). https://doi.org/10.1007/s002110050462
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DOI: https://doi.org/10.1007/s002110050462