Letter to the editor: A new criterion for Gabor frames
type: Item
instance of: scholarly article
title: Letter to the editor: A new criterion for Gabor frames
zbMATH Open document ID: 1022.42021
published in: The Journal of Fourier Analysis and Applications
publication date: 2003-06-23T00:00:00Z
full work available at URL: letter-to-the-editor-a-new-criterion-for-gabor-frames(edc8cf42-3018-4a29-9966-9cca87b86833).html
review text: The main result gives a condition for a Gabor system \(\{e^{2\pi i \beta n\cdot t}g(t-\alpha k)\}_{k,n\in \mathbb{Z}^d}\) to be a frame for \(L^2(\mathbb{R}^d)\). Assume that \(\alpha \beta\) is irrational and that \(g\) belongs to the Wiener amalgam space \(W(C,\ell^1)\), i.e., \(g\) is continuous and \( \sum_{k\in \mathbb{Z}^d} \sup_{t\in [0,1]^d}|g(t+k)|<\infty\). Let \(G\) denote the bi-infinite matrix-valued function whose \(jl\)-th entry is \(G_{jl}(t)=\sum_{k\in \mathbb{Z}^d} \overline{g}(x-l/\beta -\alpha k)g(x-j/\beta -\alpha k)\). Then \(\{e^{2\pi i \beta n\cdot t}g(t-\alpha k)\}_{k,n\in Z^d}\) is a frame if and only if there exists at least one \(\xi \in [0, \alpha]^d\) such that \(G(\xi)\) is invertible on \(\ell^2\).
Mathematics Subject Classification ID: 42C15
Mathematics Subject Classification ID: 42C40
zbMATH DE Number: 1890477
zbMATH Keywords: Gabor frames
zbMATH Keywords: Wiener amalgam space
author: Karlheinz Gröchenig
author: A. J. E. M. Janssen
MaRDI profile type: Publication
OpenAlex ID: W2022124455
DOI: 10.1007/S00041-002-0024-3
description: scientific article; zbMATH DE number 1890477