Existence of multiple refinable distributions
type: Item
instance of: scholarly article
title: Existence of multiple refinable distributions
zbMATH Open document ID: 0901.42022
author: Ding-Xuan Zhou
published in: Michigan Mathematical Journal
publication date: 1997-10-28T00:00:00Z
review text: The matrix refinement equation \[ \varphi= \sum_{\alpha\in \mathbb{Z}^s} a(\alpha)\varphi(2\cdot-\alpha)\tag{1} \] in which \(a(\alpha)\) is a sequence of \(r\times r\) complex matrices and \(\varphi= (\varphi_1,\dots,\varphi_r)^T\) is a vector of distributions on \(\mathbb{R}^s\), is studied. One of the main results reads as follows. Theorem. Let \(\{a(\alpha)\}_{\alpha\in\mathbb{Z}^s}\) be a finitely supported sequence of \(2\times 2\) complex matrices. Assume that \(2^n\) is not an eigenvalue of \(M= 2^{-s}\sum_{\alpha\in \mathbb{Z}^s}a(\alpha)\) for any \(n\in\mathbb{N}\). Then (1) has a solution \(\varphi= (\varphi_1,\varphi_2)^T\), a vector of compactly supported distributions on \(\mathbb{R}^s\) such that \(\widehat\varphi(0)\neq 0\), if and only if the matrix \(M\) has an eigenvalue 1. In this case, \(\widehat\varphi(0)\) is an eigenvector of \(M\) associated with the eigenvalue 1. In the above statement \(\widehat\varphi(0)\) denotes the Fourier transform of \(\varphi\) at the point \(0\).
Mathematics Subject Classification ID: 42C15
zbMATH DE Number: 1080677
zbMATH Keywords: multiple refinable distributions
zbMATH Keywords: wavelets
zbMATH Keywords: matrix refinement equation
reviewed by: Boris Rubin
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1029005707
OpenAlex ID: W1998722818
DOI: 10.1307/MMJ/1029005707
description: scientific article; zbMATH DE number 1080677