Admissible wavelets associated with the transform group on \(\mathbb{R}^ n\)
type: Item
instance of: scholarly article
title: Admissible wavelets associated with the transform group on \(\mathbb{R}^ n\)
zbMATH Open document ID: 0887.22010
author: J. Ning
published in: Applied Mathematics. Series B (English Edition)
publication date: 1998-04-28T00:00:00Z
review text: Let \(\Gamma \) be the set of diagonal matrices \(\text{diag}\{\rho _{1,\dots,}\rho _n\},\rho _i>0.\) Let \[ \mathbf P=\{(b,\rho):b\in {\mathbf R}^n,\rho \in \Gamma\}. \] Then \({\mathbf P}\) acts on \({\mathbf R}^n\) by \[ x\rightarrow (b,\rho)x=\rho x+b,\text{ where }x\in {\mathbf R}^n. \] The group law of \({\mathbf P}\) is defined by \[ (b,\rho)(b',\rho')=(\rho b'+b,\rho \rho '). \] \({\mathbf P}\) is a locally compact group with left Haar measure \(d\mu =\frac{ d^n\rho d^nb}{(\text{det } \rho )^2}.\) The authors define the unitary representation \(U\) of \({\mathbf P}\) on \(L^2({\mathbf R}^n)\) by \[ U(b,\rho )f(x)=(\text{det } \rho )^{-1/2}f(\rho ^{-1}(x-b)), \] where \(f(x)\in L^2({\mathbf R}^n),(b,\rho )\in {\mathbf P}\). The authors decompose \(L^2({\mathbf R}^n)\) into the direct sum of irreducible invariant closed subspaces under \(U\). The restrictions of \(U\) on these subspaces are square-integrable; they also give an admissibility condition in terms of the Fourier transform.
Mathematics Subject Classification ID: 22E27
Mathematics Subject Classification ID: 42C40
Mathematics Subject Classification ID: 32A07
Mathematics Subject Classification ID: 33C65
zbMATH DE Number: 1112221
zbMATH Keywords: wavelets
zbMATH Keywords: locally compact group
zbMATH Keywords: unitary representation
zbMATH Keywords: Fourier transform
reviewed by: Vladimir Rabinovich
MaRDI profile type: Publication
full work available at URL: s11766-997-0050-7
OpenAlex ID: W24530123
cites work: Decomposition of Hardy Functions into Square Integrable Wavelets of Constant Shape
cites work: Toeplitz and Hankel type operators on the upper half-plane
cites work: Wavelet Transform and Toeplitz-Hankel Type Operators.
cites work: Q3971548
DOI: 10.1007/S11766-997-0050-7
description: scientific article; zbMATH DE number 1112221