Stability and Linear Independence Associated with Scaling Vectors

Abstract

In this paper, we discuss stability and linear independence of the integer translates of a scaling vector $\Phi =(\phi_1,\ldots,\phi_r)^T$, which satisfies a matrix refinement equation $$\Phi(x)=\sum_{k=0}^n P_k\Phi(2x-k), $$ where $(P_k)$ is a finite matrix sequence. We call $P(z)=\frac{1}{2}\sum P_kz^k $ the symbol of $\Phi$. Stable scaling vectors often serve as generators of multiresolution analyses (MRAs) and therefore play an important role in the study of multiwavelets. Most useful MRA generators are also linearly independent.The purpose of this paper is to characterize stability and linear independence of the integer translates of a scaling vector via its symbol. A polynomial matrix P(z) is said to be two-scale similar to a polynomial matrix Q(z) if there is a polynomial matrix T(z) such that P(z)=T(z2 )Q(z)T-1 (z). This kind of factorization of P(z) is called two-scale factorization. We give a necessary and sufficient condition, in terms of two-scale factorization of the symbol, for stability and...