Properties of locally linearly independent refinable function vectors

by Zhou, Ding-Xuan

The paper considers properties of compactly supported, locally linearly independent refinable function vectors $Phi=(phi_1, ldots , phi_r)^T$, $r in NN$. In the first part of the paper, we show that the interval endpoints of the global support of $phi_{ u}$, $ u=1, ldots , r$, are special rational numbers. Moreover, in contrast with the scalar case $r=1$, we show that components $phi_{ u}$ of a locally linearly independent refinable function vector $Phi$ can have holes. In the second part of the paper we investigate the problem whether any shift-invariant space generated by a refinable function vector $Phi$ possesses a basis which is linearly independent over $(0,1)$. We show that this is not the case. Hence the result of Jia, that each finitely generated shift-invariant space possesses a globally linearly independent basis, is in a certain sense the strongest result which can be obtained.