Convergence of vector subdivision schemes in Sobolev spaces (Q1604501)

Let \(M\) be a dilation matrix, i.e. an invertible integer-valued matrix such that \(\lim_{n\to \infty }M^{-n}=0, \) and let \(a\) be a matrix-valued finitely supported function on the lattice \( {{\mathbb Z}}^{s}\), the so-called refinement mask. Further let \(A\) be the mask symbol defined by \[ A( \omega) =\frac{1}{|\det M|}\sum_{\alpha \in Z^{s}}a( \alpha) e^{-i\alpha \cdot \omega }. \] The authors start with the following interesting observation: Assume that \(M \) is isotropic, \(\varphi _{i}\), \(i=1,\dots,r\), are compactly supported functions in the Sobolev space \(W_{p}^{k}( {{\mathbb R}}^{s}) \) such that \(\widehat{\Phi } ( 0) \neq 0\) and span\(\{ \widehat{\Phi }( 2\pi \beta) :\beta \in {\mathbb Z}^{s}\} ={\mathbb C}^{r}\) . If \(\Phi =( \varphi _{1},\dots ,\varphi _{r}) \) is a solution of the refinement equation \[ \Phi =\sum_{\alpha \in {{\mathbb Z}}^{s}}a( \alpha) \Phi ( M\cdot -\alpha) \] then \(A( 0) \) satisfies the so-called \textit{eigenvalue condition of order} \(k\): \(A( 0) \) has \(1\) as a simple eigenvalue and the other eigenvalues are of modulus less than \(\rho ^{-k}\) where \(\rho \) denotes the spectral radius of the dilation matrix \(M.\) The cascade operator \(Q_{a}\) is defined by \( Q_{a}\Phi =\sum_{\alpha \in {{\mathbb Z}}^{s}}a( \alpha) \Phi ( M\cdot -\alpha). \) The aim of the paper is to discuss the subdivision scheme \(\Phi _{n}=Q_{a}\Phi _{n-1}\) (or the cascade algorithm) with respect to the \textit{Sobolev norm} under the assumption that the eigenvalue condition is satisfied. The first main result is: if \(\Phi _{0}\) is an initial vector of compactly supported functions in \(W_{p}^{k}( {{\mathbb R}}^{s}) \) and \( Q_{a}^{n}\Phi _{0}\) converges to a compactly supported function \(\Phi \) in \( W_{p}^{k}( {{\mathbb R}}^{s}) \) then the Fourier transform \(\widehat{\Phi _{0} }\) of the initial vector \(\Phi _{0}\) satisfies Strang-Fix type conditions up to the order \(k\). This class of initial vectors \(\Phi _{0}\) is denoted by \(Y_{k}.\) A similar result was obtained by \textit{Q. Chen, J. Liu} and \textit{W. Zhang} [J. Comput. Math. 20, No. 4, 363-372 (2002; Zbl 1006.65153), reviewed above]. The subdivision scheme is said to be convergent in the Sobolev space \(W_{p}^{k}( {{\mathbb R}}^{s}) \) if there exists a compactly supported function \(\Phi \) in \(W_{p}^{k}( {{\mathbb R}}^{s}) \) such that for any initial vector \(\Phi _{0}\) in the class \(Y_{k}\) the scheme \(Q_{a}^{n}\Phi _{0}\) converges to \(\Phi\). The second main result shows that the subdivision operator \(S_{a}\) associated with a mask \(a\) possesses a natural invariant subspace defined by means of the eigenvalue condition provided that the subdivision scheme \(Q_{a}^{n}\) converges in \( W_{p}^{k}( {{\mathbb R}}^{s}) .\) The third main result characterizes the convergence of a subdivison scheme in \(W_{p}^{k}( {{\mathbb R}}^{s}) \) in terms of the \(p\)-norm joint spectral radius of a finite collection of transition operators determined by the sequence \(a\) restricted to a certain invariant subspace. An application of these results is given by \textit{D. Chen} and \textit{X. Zheng} [J. Math. Anal. Appl. 268, No. 1, 41-52 (2002; Zbl 1006.65155), reviewed below].