Stationary subdivision for general scaling matrices (Q1127709)

Let \(M\in\mathbb{Z}^{s\times s}\) be an expanding scaling matrix (modulus of eigenvalues larger-than 1) and \(a=(a_\alpha)_{\alpha\in \mathbb{Z} ^s}\) a mask (sequence with finite support). Then the corresponding stationary subdivision scheme is the mapping \(S_{a,M}\colon 1_\infty(\mathbb{Z} ^s)\to 1_\infty (\mathbb{Z} ^s)\) defined by \((S_{a,M}\lambda)_\alpha=\sum_{\beta \in\mathbb{Z} ^s} a_{\alpha-M \beta}\lambda_\beta\). The subdivision scheme is said to be uniformly convergent if there exists a function \(f_\lambda \in C({\mathbb{R}}^s)\) such that \(\lim_{m\to\infty} \|(S^m_{a,M}\lambda)-f_{\lambda(M^{-m} \bullet)}\|_{ 1_\infty(\mathbb{Z} ^2)}=0\). Such subdivision schemes are important for geometric design and multivariate wavelet analysis. Most of the results for the classical case \(M=2I\) are easily generalized. However, for the convergence, which is studied in this paper, some specific algebraic tools need to be developed in the case of a general \(M\). The main theorem gives a characterization of convergent subdivision schemes with general \(M\) and compactly supported mask.