Study of Linear Independence and Accuracy of Scaling Vectors via Two-scale Factors

A scaling vector = (1 ; ; r) T is a compactly supported and vector-valued distribution vector that satisses a matrix reenement equation (x) = P P k (2x ? k); where (P k) is a nite matrix sequence. We call P (z) = 1 2 P P k z k the symbol of. A symbol P (z) is said two-scale similar to a polynomial matrix Q(z) if there is an invertible polynomial matrix T (z) that satisses P (z) = T (z 2)Q(z)T ?1 (z): T (z) is called a two-scale factor of P (z). In this paper we characterize linear independence and accuracy associated with a scaling vector via the two-scale factor of its symbol. The necessary and suucient conditions for either linear independence or for accuracy are given. The relation between them are also revealed. 1. INTRODUCTION In this paper, we study linear independence and accuracy associated with scaling vectors. A compactly supported distribution vector