Multivariate symmetric refinable functions and function vectors (Q2819174)

Multivariate symmetric refinable functions and refinable function vectors are of interest in both theory and applications. Let \(M\) be a \(d\times d\) integer matrix such that all the eigenvalues of \(M\) are greater than one in modulus. Let \(H\) be a symmetry group that is compatible with \(M\) (i.e., \(M^{-1}EM\in H\) for all \(E\in H\)) and \(c\in \mathbb{R}^d\) be an appropriate symmetry center for \(H\). For any positive integer \(n\), Theorem~8 constructs scalar \(H\)-symmetric masks \(m_0\) with respect to the center \(c\) such that \(m_0\) has sum rules of order \(n\). The whole class of scalar \(H\)-symmetric masks with sum rules of order \(n\) is described in Theorem~11. Similar corresponding results for refinable function vectors and matrix masks are presented in Section~4. Several interesting examples of scalar symmetric masks and matrix symmetric masks are presented in Section~5 for symmetric scalar refinable functions and symmetric refinable function vectors.