Number systems, \(\alpha\)-splines and refinement (Q704188)

The author introduces a family of \(\alpha\)-splines, which are convolution powers of the \(\alpha\)-basis functions \(B_{0,\xi}\) associated with some lattice in \(\mathbb C\), defined as the functions which satisfy the refinement equation: \[ B_{0,\xi} (z) = \sum_{j=1}^{| \alpha| ^2} B_{0,\xi}(\alpha z - n_j), \] for a.e. \(z \in {\mathbb C}\). Here, \(n_j\) are elements of the lattice satisfying some additional requirements and \(\alpha \in {\mathbb C}\) preserves the lattice. Several examples illustrating the theory are provided.