BMO, boundedness of affine operators, and frames
type: Item
instance of: scholarly article
title: BMO, boundedness of affine operators, and frames
zbMATH Open document ID: 1069.42023
published in: Applied and Computational Harmonic Analysis
publication date: 2005-02-23T00:00:00Z
review text: For real numbers \(r>1, s>0\), define the dilation operator \(D_rf(x)=r^{1/2}f(rx)\) and the translation operator \(T_sf(x)=f(x-s)\). For \(\phi,\psi \in L^2(R)\), define (formally) the operator \(D_{\phi,\psi}^{r,s}f= \sum_{k,l\in Z} \langle f,D_r^kT_s^l \phi\rangle D_r^kT_s^l \psi\). Several conditions for \(D_{\phi,\psi}^{r,s}\) being bounded on \(l^2(R)\) are presented, and related to the vanishing moments of \(\phi\) and \(\psi\). For example, if \(\phi\) and \(\psi\) are smooth and have a vanishing moment, then the associated operator is automatically bounded; this might no longer hold if only one of the functions has a vanishing moment. The results in the paper are obtained via the theory for singular integral operators.
Mathematics Subject Classification ID: 42C40
Mathematics Subject Classification ID: 40A30
zbMATH DE Number: 2137818
zbMATH Keywords: singular integral operator
zbMATH Keywords: affine system
zbMATH Keywords: wavelets
zbMATH Keywords: Bessel condition
zbMATH Keywords: BMO
zbMATH Keywords: frames
author: John E. Gilbert
author: Jeffrey A. Hogan
author: Joseph Lakey
MaRDI profile type: Publication
full work available at URL: j.acha.2004.05.005
OpenAlex ID: W1976862588
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DOI: 10.1016/J.ACHA.2004.05.005
description: scientific article; zbMATH DE number 2137818