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Register a new userGabor frame characterisations of generalised modulation spaces
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We obtain Gabor frame characterisations of modulation spaces defined via a class of translation-modulation invariant Banach spaces of distributions that was recently introduced in $[10]$. We show that these spaces admit an atomic decomposition through Gabor expansions and that they are characterised by summability properties of their Gabor coefficients. Furthermore, we construct a large space of admissible windows. This generalises several fundamental results for the classical modulation spaces $ M^{p,q}_{w}$. Due to the absence of solidity assumptions on the Banach spaces defining these modulation spaces, the methods used for the spaces $M^{p,q}_{w}$ (or, more generally, in coorbit space theory) fail in our setting and we develop here a new approach based on the twisted convolution.
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Given a non-quasianalytic subadditive weight function $omega$ we consider the weighted Schwartz space $mathcal{S}_omega$ and the short-time Fourier transform on $mathcal{S}_omega$, $mathcal{S}_omega$ and on the related modulation spaces with exponential weights. In this setting we define the $omega$-wave front set $WF_omega(u)$ and the Gabor $omega$-wave front set $WF^G_omega(u)$ of $uinmathcal{S}_{omega}$, and we prove that they coincide. Finally we look at applications of this wave front set for operators of differential and pseudo-differential type.
In this paper, we consider the trace theorem for modulation spaces, alpha modulation spaces and Besov spaces. For the modulation space, we obtain the sharp results.
We study the phase reconstruction of signals $f$ belonging to complex Gaussian shift-invariant spaces $V^infty(varphi)$ from spectrogram measurements $|mathcal{G}f(X)|$ where $mathcal{G}$ is the Gabor transform and $X subseteq mathbb{R}^2$. An explicit reconstruction formula will demonstrate that such signals can be recovered from measurements located on parallel lines in the time-frequency plane by means of a Riesz basis expansion. Moreover, connectedness assumptions on $|f|$ result in stability estimates in the situation where one aims to reconstruct $f$ on compact intervals. Driven by a recent observation that signals in Gaussian shift-invariant spaces are determined by lattice measurements [Grohs, P., Liehr, L., Injectivity of Gabor phase retrieval from lattice measurements, arXiv:2008.07238] we prove a sampling result on the stable approximation from finitely many spectrogram samples. The resulting algorithm provides a non-iterative, provably stable and convergent approximation technique. In addition, it constitutes a method of approximating signals in function spaces beyond $V^infty(varphi)$, such as Paley-Wiener spaces.
We give sufficient conditions for compactness of localization operators on modulation spaces $textbf{M}^{p,q}_{m_{lambda}}( mathbb{R}^{d})$ of $omega$-tempered distributions whose short-time Fourier transform is in the weighted mixed space $L^{p,q}_{m_lambda}$ for $m_lambda(x)=e^{lambdaomega(x)}$.
This paper is devoted to give several characterizations on a more general level for the boundedness of $tau$-Wigner distributions acting from weighted modulation spaces to weighted modulation and Wiener amalgam spaces. As applications, sharp exponents are obtained for the boundedness of $tau$-Wigner distributions on modulation spaces with power weights. We also recapture the main theorems of Wigner distribution obtained in cite{CorderoNicola2018IMRNI,Cordero2020a}. As consequences, the characterizations of the boundedness on weighted modulation spaces of several types of pseudodifferential operators are established. In particular, we give the sharp exponents for the boundedness of pseudodifferential operators with symbols in Sj{o}strands class and the corresponding Wiener amalgam spaces.
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