Properties of locally linearly independent refinable function vectors
Let \(\Phi= (\phi_1,\dots,\phi_r)^T\) be a vector of compactly supported integrable functions on \(\mathbb{R}\) such that \[ \Phi(t)= \sum_{k\in\mathbb{Z}} A(k)\Phi(2t- k)\quad\text{for all }t\in \mathbb{R}, \] where \(\{A(k)\}\) is a finitely supported sequence of \((r\times r)\)-matrices. Under the assumption that \(\Phi\) is locally linearly independent, the following results are established: (1) the global supports of the components of \(\Phi\) start and end with special rational numbers; (2) in contrast with the scalar case, the components of \(\Phi\) may possess a hole in their global support when \(r\geq 2\); (3) if \(A(k)= 0\) for \(k< 0\) and \(k> N\), and if \(A(0)\) and \(A(N)\) do not contain zero rows, then all nonzero components of \(\Phi\) have the global support \([0,N]\) and, in particular, they have no holes. Furthermore, in the case \(r= 2\) there are given two interesting globally linearly independent functions \(\Phi\): (1) a function which is linearly dependent over \((0,1)\) and for which the shift-invariant space generated by \(\Phi\) does not possess a linearly independent basis over \((0,1)\); (2) a function which is linearly dependent over any finite interval.
scientific article; zbMATH DE number 1924729