Sampling measures for the Gabor transform
type: Item
instance of: scholarly article
title: Sampling measures for the Gabor transform
zbMATH Open document ID: 1331.42032
author: Gerard Ascensi
published in: Journal of Approximation Theory
publication date: 2015-09-28T00:00:00Z
review text: The aim of this paper is to study sampling measures for model spaces of the Gabor transform. The main interest for the study of these measures comes from their relation with the theory of Gabor frames. To formulate the main results of the article we need to introduce some notation. Let \(T_x\) and \(M_y\) be the translation and modulation operators respectively. For \(\varphi\in L^2({\mathbb R}^d)\) with \(\|\varphi\|_2 =1,\) the Gabor transform of \(f\in L^2({\mathbb R}^d)\) is defined by \[ V_\varphi f(x,y) = \left<f, M_yT_x\varphi\right> = \int_{{\mathbb R}^d} f(t)\overline{e^{2\pi i yt}\varphi(t-x)}\;dt. \] For a fixed \(\varphi\in L^2({\mathbb R}^d)\) such that \(V_\varphi\varphi\in L^1({\mathbb R}^{2d}),\) the modulation class \(M^p({\mathbb R}^d)\) consists of those distributions \(g\) such that \(V_\varphi g\in L^p({\mathbb R}^{2d}).\) This definition does not depend on \(\varphi,\) and the model space \(G^p_\varphi\) is the image of \(M^p({\mathbb R}^d)\) under \(V_\varphi.\) Then \(G^2_\varphi\) is a reproducing kernel Hilbert space with reproducing kernel \(k_\varphi.\) Now we can introduce the concept under study. A Borel measure \(\mu\) in \({\mathbb R}^{2d}\) is a sampling measure for \(G^p_\varphi\) (\(1\leq p < \infty\)) if there exist constants \(A, B > 0\) such that for all \(f\in G^p_\varphi\) \[ A\| f\|_p^p\leq \int_{{\mathbb R}^{2d}}\left|f(z)\right|^p\;d\mu(z) \leq B\| f\|_p^p. \] Section 3 contains a necessary and a sufficient condition for sampling measures in terms of uniqueness sets of certain spaces. To be more precise we need to consider the set \(W_\mu\) consisting of all weak limits of translations of the measure \(\mu.\) Then Theorem 14 reads as follows. Let \(\mu\) be a Bessel measure and \(\varphi\in {\mathcal M}^1\) and \(1\leq p < \infty.\) If the support of any measure in \(W_\mu\) is a uniqueness set for \(G^p_\varphi(\vartheta dm)\) then \(\mu\) is a sampling measure for \(G^p\varphi.\) The condition \(\varphi\in {\mathcal M}^1\…
reviewed by: Antonio Galbis
Mathematics Subject Classification ID: 42C15
Mathematics Subject Classification ID: 42B35
zbMATH DE Number: 6485846
zbMATH Keywords: short-time Fourier transform
zbMATH Keywords: modulation space
zbMATH Keywords: sampling sets
zbMATH Keywords: sampling measures
MaRDI profile type: Publication
full work available at URL: j.jat.2015.04.002
OpenAlex ID: W330390204
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DOI: 10.1016/J.JAT.2015.04.002
description: scientific article; zbMATH DE number 6485846