Convergence of cascade algorithms in Sobolev spaces associated with inhomogeneous refinement equations
The paper is concerned with the solutions \(\varphi= (\varphi_1,\dots, \varphi_r)'\) of multivariate refinement equations \[ \varphi(x)= \sum_{\alpha\in\mathbb{Z}^s} a(\alpha)\varphi(2x- \alpha)+ g(x)\qquad (x\in\mathbb{R}^s), \] where \(g:= (g_1,\dots, g_r)'\) and \(a\) is a finitely supported refinement mask such that \(a(\alpha)\in \mathbb{C}^{r\times r}\). The author gives a characterization for the strong convergence of the cascade algorithm \[ \varphi^{(n)}(x)= \sum_{\alpha\in \mathbb{Z}^s} a(\alpha)\varphi^{(n-1)}(2x- \alpha)+ g(x)\qquad (x\in \mathbb{R}^s) \] in the Sobolev space \(W^k_2(\mathbb{R}^s)\) \((k\in\mathbb{N})\) in terms of the refinement mask \(a\), the inhomogeneous term \(g\) and the initial function vector \(\varphi^{(0)}\).
scientific article; zbMATH DE number 1455763