Sampling measures for the Gabor transform
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 \(\varphi\in {\mathcal M}^1\) means that the reproducing kernel \(k_\varphi\) is controlled in a certain way by a weight \(\vartheta\) (the precise control condition as well as the definition of the weighted \(G^p_\varphi(\vartheta dm)\) are contained in Section 2). A partial converse is given by Proposition 15, let say, if \(\mu\) is a sampling measure for \(G^p\varphi\) then the support of any measure in \(W_\mu\) is a uniqueness set for \(G^p_\varphi.\) Section 4 contains a stability result for the sampling property with respect to certain perturbations on the support of a measure. The obtained results are used in Section 5 to characterize sampling measures in terms of sampling discrete sets. Let \(\mu\) be a Bessel measure and denote by \(\mu_\varepsilon\) the \(\varepsilon\)-reduction of \(\mu,\) which is carefully analyzed in Section 4. Then \(\mu\) is a sampling measure if and only if there exists \(\varepsilon > 0\) and a separated set \(\Lambda\) such that \(\lambda \subseteq \text{supp}\;\mu_\varepsilon\) and \(\sum_{\lambda\in\Lambda}\delta_\lambda\) is a sampling measure, with \(\delta_x\) the Dirac measure in \(x.\) This is the content of Theorem 22. The final Section 6 contains some applications to the study of planar sets of sampling and annihilating sets for the short time Fourier transform.
scientific article; zbMATH DE number 6485846
scientific article