Properties of locally linearly independent refinable function vectors Original Research Article

The paper considers properties of compactly supported, locally linearly independent refinable function vectors Φ=(φ1,…,φr)T, r∈N. In the first part of the paper, we show that the interval endpoints of the global support of φν, ν=1,…,r, are special rational numbers. Moreover, in contrast with the scalar case r=1, we show that components φν of a locally linearly independent refinable function vector Φ 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 Φ 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.