Two-scale homogeneous functions in wavelet analysis.
type: Item
instance of: scholarly article
title: Two-scale homogeneous functions in wavelet analysis.
zbMATH Open document ID: 1040.42040
author: Ding-Xuan Zhou
published in: The Journal of Fourier Analysis and Applications
publication date: 2002-01-01T00:00:00Z
review text: In this paper two-scale homogeneous functions, satisfying \(h(x) = \lambda \cdot h(2x)\) for some constant \(\lambda\), are considered. Let \(\phi = (\phi_1, \ldots , \phi_r)^T\) be a refinable vector of functions, i.e., \[ \phi (x) = \sum_{k\in {\mathbb Z}} a(k) \phi (2x-k), \] where \(a := (a(k))_{k\in {\mathbb Z}}\) is a sequence of \(r \times r\) matrices. The author investigates two-scale homogeneous functions that are contained in the shift invariant space \(S(\phi)\) generated by the integer translates of \(\phi_1, \ldots , \phi_r.\) This study leads to some interesting conclusions. There is a one-to-one correspondence between two-scale homogeneous functions of order \(\lambda \) and left eigenvectors of \((a(2j-k))^{N-1}_{j,k=0}\) to the eigenvalue \(\lambda\). It can now be shown that among all refinable function vectors with fixed support the B-spline vector with multiple knots yields optimal accuracy. Further, the subdivision operator related to \((a(k))_{k \in {\mathbb Z}}\) is shown to have no eigenvalues in \(l_p({\mathbb Z})\). Finally, the reconstruction of \(\phi\) from two-scale homogeneous functions is considered.
Mathematics Subject Classification ID: 42C40
Mathematics Subject Classification ID: 41A25
Mathematics Subject Classification ID: 41A15
Mathematics Subject Classification ID: 65F15
zbMATH DE Number: 1921137
zbMATH Keywords: two-scale homogeneous function
zbMATH Keywords: refinable function
zbMATH Keywords: accuracy
zbMATH Keywords: subdivision operator
zbMATH Keywords: B-splines
reviewed by: Gerlind Plonka
MaRDI profile type: Publication
full work available at URL: s00041-002-0027-0
OpenAlex ID: W2013215404
DOI: 10.1007/S00041-002-0027-0
description: scientific article; zbMATH DE number 1921137