Stability implies convergence of cascade algorithms in Sobolev space (Q1604226)

The main result of the paper states the following: suppose that \(\Phi =(\varphi _{1},\dots ,\varphi _{r})\) is a vector of compactly supported functions in the Sobolev space \(W_{p}^{k}( {{\mathbb R}}^{s}) \) such that \(\Phi \) satisfies the refinement equation with respect to a finitely supported refinement mask \(a\) and an isotropic dilation matrix \(M.\) If the shifts of \( \varphi _{1},\dots ,\varphi _{r}\) are stable then the cascade algorithm \( Q_{a}^{n}\Phi _{0}\) associated with \(a\) and \(M\) converges to \(\Phi \) in the Sobolev norm for any initial vector \(\Phi _{0}\) from a feasible set \(Y_{k}.\) The proof depends on results by \textit{D. Chen, R. Jia} and \textit{S. D. Riemenschneider} [Appl. Comput. Harmon. Anal. 12, No. 1, 128-149 (2002; Zbl 1006.65154), reviewed above].