Stability and Linear Independence Associated with Scaling Vectors

Jianzhong Wang
1998 SIAM Journal on Mathematical Analysis  
In this paper, we discuss stability and linear independence of the integer translates of a scaling vector Φ = (φ 1 , · · · , φ r ) T , which satisfies a matrix refinement equation where (P k ) is a finite matrix sequence. We call P (z) = 1 2 P k z k the symbol of Φ. Stable scaling vectors often serve as generators of multiresolution analyses (MRA) and therefore play an important role in the study of multiwavelets. Most useful MRA generators are also linearly independent. The purpose of this
more » ... r 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 (z 2 )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 linear independence of the integer translates of a scaling vector. where the matrix sequence (P k ) k∈Z is called a mask of Φ. Taking the Fourier transform of both sides of equation (1.1), we obtain where P (z) := 1 2 P k z k is a symbol of Φ. We callΦ(0) the moment (of order 0) of Φ sinceΦ(0) = R Φ(x)dx. WhenΦ(0) = 0, we call Φ a zero-moment scaling vector. We will characterize stability and linear independence of the integer translates of a scaling vector via a special factorization of its symbol. Our study of scaling vectors is based on shift-invariant spaces. Hence, we first introduce some notions and results in the theory of shift-invariant spaces. Let S be a linear space of distributions on R. We say that S is shift-invariant if f ∈ S =⇒ f (· − j) ∈ S ∀j ∈ Z.
doi:10.1137/s003614109630330x fatcat:mbja5vwumrahtku2ni6cchivi4