Connections between the support and linear independence of refinable distributions (Q1266093)

A compactly supported distribution \(\phi\in \mathcal E'(\mathbb R)\) is \textit{refinable} if it satisfies a refinement equation such as \(\phi(\cdot)=\sum_{j=0}^M p_j\phi(N\cdot-j)\) where \(p_0,\dots,p_M\) are real numbers and \(N\) is an integer dilation factor. The support of \(\phi\) is the complement of the open set \(\mathcal O\) such that \(\langle \phi, f\rangle =0\) whenever \(f\) is a test function supported in \(\mathcal O\). One can define the space \(V_\phi=\text{span}\{\phi(\cdot -k)\}_{k\in \mathbb Z}\). One says that \(\{\phi(\cdot -k)\}\) is globally linearly independent if the coefficients of \(\psi\in V_\phi\) are uniquely determined, and that \(\{\phi(\cdot -k)\}\) is locally linearly independent if, whenever \(\psi\in V_\phi\) vanishes on an interval, the coefficients of those \(\phi(\cdot -k)\) whose supports overlap that interval must equal zero. In this paper the authors prove that if \(\phi\) satisfies a refinement equation and the integer shifts of \(\phi\) are globally linearly independent then, in fact, they are locally linearly independent. Moreover, the support of \(\phi\) is then an interval: it has no gaps. Although \(\phi\) may not generate a multiresolution analysis in the usual sense, the nestedness of distribution spaces generated by dilates and shifts of \(\phi\) plays a fundamental role in the analysis.