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Dilation of the Weyl symbol and Balian-low theorem. (English) Zbl 1298.47059

The main result (Thm. 1.1) states that the property of invertibility in a class of pseudodifferential operators with Weyl symbol in the modulation space \(M^{\infty,1}(\mathbb{R}^{2n})\) is preserved under small dilation of the Weyl symbol. The precise formulation is as follows. Let \(A_\sigma\) denote the pseudodifferential operator with Weyl symbol \(\sigma \in M^{\infty,1}(\mathbb{R}^{2n}):=\int\| V_\varphi(\cdot, \omega) \|_{L^\infty}\,d \omega\), where \(V_\varphi\) denotes the short-time Fourier transform with the Gaussian \(\varphi\) as window. For \(\rho \in GL(2n,\mathbb{R})\), define the dilation operator by \(D_\rho f(z)=f(\rho^{-1}z)\). Then the set \(\Sigma=\{(\sigma,\rho)\in M^{\infty,1}(\mathbb{R}^{2n}) \times GL(2n,\mathbb{R}) \, : \, A_{D_\rho \sigma} \text{ is invertible} \}\) is an open set in \(M^{\infty,1}(\mathbb{R}^{2n}) \times\mathrm{GL}(2n,\mathbb{R})\).
The proof of Thm. 1.1 is based on Lemma 3.1 that gives sufficient conditions under which the set \(F=\{(\sigma,g)\in M \times G: K_g \sigma \text{ is invertible}\}\) is open; here, \(M\) is a unital Banach algebra and \(G\) a topological group acting on \(M\) by a family of bounded linear operators \((K_g)_{g \in G}\). In particular, the authors show that the family of dilations \((D_\rho)_\rho\) on the Banach algebra \((M^{\infty,1},\sharp)\), where \(\sharp\) is the usual twisted product of Weyl symbols, satisfies the assumptions of Lemma 3.1.
Thm. 1.1 has a number of implications for Gabor systems in time-frequency analysis. A Gabor system \(G(g,\Lambda)\) is formed by the time-frequency shifts of some function \(g \in L^2(\mathbb{R}^n)\) along a discrete set \(\Lambda\) of time-frequency points in \(\mathbb{R}^{2n}\): \[ G(g,\Lambda) = \{g_\lambda : \lambda \in \Lambda \}, \quad g_\lambda(t):= e^{2\pi i \omega \cdot t} g(t-x), \quad \lambda=(x,\omega) \in \mathbb{R}^{2n}. \] The Gabor system is said to be regular (or uniform) if \(\Lambda\) is a full-rank lattice in \(\mathbb{R}^{2n}\), and irregular (or non-uniform) otherwise. Thm. 1.1 implies the stability of Gabor frames under small dilations \(\rho\Lambda\) of the time-frequency set \(\Lambda\) (Thm. 1.3), where \(\rho\) is in a neighborhood of \(I_{2n}\); Thm. 1.3 extends results from [H. G. Feichtinger and N. Kaiblinger, Trans. Am. Math. Soc. 356, No. 5, 2001–2023 (2004; Zbl 1033.42033)] from the case of regular Gabor systems to irregular Gabor systems. Thm. 1.1 also implies a Balian-Low theorem (Thm. 1.5) for irregular Gabor systems with window \(g\) in the Feichtinger algebra \(M^{1,1}(\mathbb{R}^n)\), showing the non-existence of irregular Gabor frames at critical density. Finally, the authors discuss implications in the theory of sampling and interpolation in the Bargmann-Fork space.
Dilations of the set \(\Lambda\) considered in this paper are linear deformations; for recent results on (non-linear) Lipschitz deformation, see [K. Gröchenig, J. Ortega-Cerda and J. L. Romero, “Deformation of Gabor systems”, Preprint, arxiv:1311.3861].

MSC:

47G30 Pseudodifferential operators
42C15 General harmonic expansions, frames
81S30 Phase-space methods including Wigner distributions, etc. applied to problems in quantum mechanics

Citations:

Zbl 1033.42033
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References:

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