Finite-dimensional approximation of the inverse frame operator
type: Item
instance of: scholarly article
title: Finite-dimensional approximation of the inverse frame operator
zbMATH Open document ID: 0961.42020
DOI: 10.1007/BF02510119
author: Ole Christensen
published in: The Journal of Fourier Analysis and Applications
publication date: 2001-05-06T00:00:00Z
full work available at URL: https://eudml.org/doc/59632
review text: A frame in a Hilbert space \(H\) allows every element of \(H\) to be written as a linear combination of the frame elements, with coefficients called frame coefficients. Calculations of these coefficients and many other situations where frames occur, require knowledge of the inverse frame operator, but it is usually difficult to invert the frame operator if the underlying Hilbert space is infinite--dimensional. In this paper the author introduces a method of approximation of the frame operator using finite subsets of the frame. In particular, this allows one to approximate the frame coefficients (even in the \(\ell^2\) sense) using finite--dimensional linear algebra. The author also shows that the method can be simplified in the important cases of Weil--Heisenberg frames and wavelet frames.
reviewed by: Richard A. Zalik
Mathematics Subject Classification ID: 42C40
Mathematics Subject Classification ID: 41A35
zbMATH DE Number: 1445627
zbMATH Keywords: frames
zbMATH Keywords: approximation of the inverse frame operator
zbMATH Keywords: Weil-Heisenberg frame
zbMATH Keywords: wavelet frame
MaRDI profile type: Publication
cites work: The theory of multiresolution analysis frames and applications to filter banks
cites work: Q4877145
cites work: Q4240902
cites work: Frames and the projection method
cites work: Frames and pseudo-inverses
cites work: Frames of exponentials: Lower frame bounds for finite subfamilies and approximation of the inverse frame operator
cites work: Q4003152
cites work: Q4003151
cites work: Ten Lectures on Wavelets
cites work: The wavelet transform, time-frequency localization and signal analysis
cites work: Continuous and Discrete Wavelet Transforms
cites work: Linear independence of time-frequency translates
cites work: Von Neumann algebras and linear independence of translates
description: scientific article; zbMATH DE number 1445627