Some characterizations for box spline wavelets and linear Diophantine equations
For univariate compactly supported distributions satisfying a refinement equation \[ \varphi=\sum_{\alpha\in {\mathbb Z}}b_\alpha \varphi (k \cdot -\alpha) \] with \(2\leq k\in{\mathbb N}\), \textit{R.-Q. Jia} and \textit{J. Wang} [Proc. Am. Math. Soc. 117, No. 4, 1115-1124 (1993; Zbl 0770.42018)] presented necessary and sufficient conditions for linear independence of integer translates of \(\varphi\) in terms of the refinement mask of \(\varphi\). In the multivariate case these conditions remain to be necessary. However, the author gives examples of box splines, where the sufficiency of these conditions does not hold any longer. On the other hand, if the box spline \(M_{\Xi}\) in \({\mathbb R}^s\) satisfies rank \(\Xi=s\), then a direct extension of the criterion for linear independence of refinable distributions holds. This approach hence opens a new way of deriving the linear independence of translates of box splines by means of their refinement masks.
scientific article; zbMATH DE number 1309891
scientific article