The spectral function of shift-invariant spaces
type: Item
instance of: scholarly article
title: The spectral function of shift-invariant spaces
zbMATH Open document ID: 1059.42021
author: Marcin Bownik
author: Ziemowit Rzeszotnik
published in: Michigan Mathematical Journal
publication date: 2003-10-27T00:00:00Z
review text: Let \( V \) be a shift-invariant closed subspace of \( L^2 (\mathbb R^n)\) generated by \(\Phi \subset L^2 (\mathbb R^n )\), \[ S(\Phi ) = \overline{\text{span}} \{\varphi(\cdot - k) : \varphi \in \Phi, \; k \in\mathbb Z^n\}. \] If \(\{ \varphi(\cdot - k) : \varphi \in \Phi , k \in \mathbb Z^n\}\) forms a tight frame of \( V= S(\Phi ) \) with constant 1, then the spectral function \( \sigma_\nu \) is given by \[ \sigma_\nu (\xi) = \sum_{\varphi \in \Phi } \; |\widehat \varphi (\xi )|^2. \] In the paper, the spectral function is systematically studied. It is applied to characterize the approximation order of \( S(\Phi )\), and to give a complete characterization of dimension functions. Further, an analogue of the Calderón reproducing formula for generalized MRAs is given and an elementary proof of Rieffel's incompleteness theorem for Garbor systems is presented.
Mathematics Subject Classification ID: 42C15
Mathematics Subject Classification ID: 42C40
Mathematics Subject Classification ID: 41A15
Mathematics Subject Classification ID: 42B10
zbMATH DE Number: 1997320
zbMATH Keywords: generalized multiresolution analysis
zbMATH Keywords: spectral function
zbMATH Keywords: shift-invariant space
zbMATH Keywords: dimension function
zbMATH Keywords: range function
zbMATH Keywords: multiplicity function
zbMATH Keywords: tight frames
zbMATH Keywords: incompleteness
zbMATH Keywords: Garbor systems
reviewed by: Gerlind Plonka
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1060013204
OpenAlex ID: W2130890381
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DOI: 10.1307/MMJ/1060013204
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description: scientific article; zbMATH DE number 1997320