Properties of locally linearly independent refinable function vectors

The paper considers properties of compactly supported, locally linearly independent refinable function vectors <f>Φ=(φ1,…,φr)T</f>, <f>r∈N</f>. In the first part of the paper, we show that the interval endpoints of the global support of <f>φν</f>, <f>ν=1,…,r</f>, are special rational numbers. Moreover, in contrast with the scalar case <f>r=1</f>, we show that components <f>φν</f> of a locally linearly independent refinable function vector <f>Φ</f> 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 <f>Φ</f> possesses a basis which is linearly independent over <f>(0,1)</f>. 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. [Copyright &y& Elsevier]