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   -> Volume 5, Issue 4


Preprint: Stability and Linear Independence of Scaling Vectors
 
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"Jianzhong Wang" (mth_jxw@shsu.edu)
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PostPosted: Wed Dec 04, 2002 9:52 am    
Subject: Preprint: Stability and Linear Independence of Scaling Vectors
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Preprint: Stability and Linear Independence of Scaling Vectors

Preprint: Stability and Linear Independence Associated with Scaling
Vectors.

Author: Jianzhong Wang
Department of Mathematical and Information Sciences
Sam Houston StateUniversity
Huntsville, TX 77341.
E-mail : jwang@galois.shsu.edu

Abstract: In this paper, we discuss stability and linear independence
of the integer translates of a scaling vector $Phi =(phi _{1,}cdots
,phi _r)^T$ which satisfies a matrix refinement equation [Phi
(x)=sum_{k=0}^nP_kPhi (2x-k),] where $(P_k)$ is a finite matrix
sequence. We call $P(z)= (1/2)sum P_kz^k$ the symbol of $Phi .$
Stable scaling vectors often serve as generators of multiresolution
analyses (MRA) and therefore play an important role in the study of
multiwavlets. Most of useful MRA generators are also linearly
independent. The purpose of this paper is to characterize stability
and linearly 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 an
invertible 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.

If want a copy of the preprint, send me an e-mail.
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