This paper addresses the problem of identifying a linear time-varying (LTV) system characterized by a (possibly infinite) discrete set of delay-Doppler shifts without a lattice (or other "geometry-discretizing&qu...
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This paper addresses the problem of identifying a linear time-varying (LTV) system characterized by a (possibly infinite) discrete set of delay-Doppler shifts without a lattice (or other "geometry-discretizing") constraint on the support set. Concretely, we show that a class of such LTV systems is identifiable whenever the upper uniform Beurling density of the delay-Doppler support sets, measured "uniformly over the class", is strictly less than 1/2. The proof of this result reveals an interesting relation between LTV system identification and interpolation in the Bargmann-Fock space. Moreover, we show that the density condition we obtain is also necessary for classes of systems invariant under time-frequency shifts and closed under a natural topology on the support sets. We furthermore find that identifiability guarantees robust recovery of the delay-Doppler support set, as well as the weights of the individual delay-Doppler shifts, both in the sense of asymptotically vanishing reconstruction error for vanishing measurement error.
We define Wiener amalgam spaces of (quasi)analytic ultradistributions whose local components belong to a general class of translation and modulation invariant Banach spaces of ultradistributions and their global compo...
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We define Wiener amalgam spaces of (quasi)analytic ultradistributions whose local components belong to a general class of translation and modulation invariant Banach spaces of ultradistributions and their global components are either weighted L-p or weighted C-0 spaces. We provide a discrete characterisation via so called uniformly concentrated partitions of unity. Finally, we study the complex interpolation method and we identify the strong duals for most of these Wiener amalgam spaces. (c) 2022 Elsevier Inc. All rights reserved.
We deduce continuity and Schatten-von Neumann properties for operators with matrices satisfying mixed quasi-norm estimates with Lebesgue and Schatten parameters in (0,infinity]. We use these results to deduce continui...
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We deduce continuity and Schatten-von Neumann properties for operators with matrices satisfying mixed quasi-norm estimates with Lebesgue and Schatten parameters in (0,infinity]. We use these results to deduce continuity and Schatten-von Neumann properties for pseudo-differential operators with symbols in quasi-Banach modulation spaces, or in appropriate Hormander classes.
We produce a finite time blow-up solution for nonlinear fractional heat equation (partial derivative(t)u + (-Delta)(beta /2) u= u(k)) in modulation and Fourier amalgam spaces on the torus T-d and the Euclidean space R...
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We produce a finite time blow-up solution for nonlinear fractional heat equation (partial derivative(t)u + (-Delta)(beta /2) u= u(k)) in modulation and Fourier amalgam spaces on the torus T-d and the Euclidean space R-d. This complements several known local and small data global well-posedness results in modulation spaces on R-d. Our method of proof rely on the formal solution of the equation. This method should be further applied to other non-linear evolution equations.
We consider Schrodinger equations with real-valued smooth Hamiltonians, and non-smooth bounded pseudo-differential potentials, whose symbols may not even be differentiable. The well-posedness of the Cauchy problem is ...
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We consider Schrodinger equations with real-valued smooth Hamiltonians, and non-smooth bounded pseudo-differential potentials, whose symbols may not even be differentiable. The well-posedness of the Cauchy problem is proved in the frame of the modulation spaces, and results of micro-local propagation of singularities are given in terms of Gabor wave front sets.
We consider fractional Hartree and cubic nonlinear Schrodin-ger equations on Euclidean space R(d )and on torus T-d. We establish norm inflation (a stronger phenomena than standard ill-posedness) at every initial data ...
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We consider fractional Hartree and cubic nonlinear Schrodin-ger equations on Euclidean space R(d )and on torus T-d. We establish norm inflation (a stronger phenomena than standard ill-posedness) at every initial data in Fourier amalgam spaces with negative regularity. In particular, these spaces include Fourier-Lebesgue, modulation and Sobolev spaces. We further show that this can be even worse by exhibiting norm inflation with an infinite loss of regularity. To establish these phenomena, we employ a Fourier analytic approach and introduce new resonant sets corresponding to the fractional dispersion (-Delta)(alpha/2). In particular, when dispersion index alpha is large enough, we obtain norm inflation above scaling critical regularity in some of these spaces. It turns out that our approach could treat both equations (Hartree and power -type NLS) in a unified manner. The method should also work for a broader range of nonlinear equations with Hartree-type nonlinearity.(c) 2023 Elsevier Inc. All rights reserved.
This study devotes to Heisenberg's uncertainty principles for Fourier integral operators of types I and II with function-variable symbols, i.e., the symbol s ?S-1?vs(8) (R-N) of the type I Fourier integral operato...
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This study devotes to Heisenberg's uncertainty principles for Fourier integral operators of types I and II with function-variable symbols, i.e., the symbol s ?S-1?vs(8) (R-N) of the type I Fourier integral operator is only w-dependent and the symbol t ?S-1?vs(8)(R-N) of the type II Fourier integral operator is only y-dependent. These two special Fourier integral operators are abbreviated as the FIO-I-FV and FIO-II-FV, respectively. We disclose an equivalence relation between the FIO-I-FV and the classical metaplec-tic transform, as well as the FIO-II-FV and the metaplectic transform, based on which we employ various versions of Heisenberg's uncertainty principles for the metaplectic transform, ranging from the general metaplectic transform of real-valued functions to some specific (e.g., the orthogonal, the orthonormal, the minimum eigen-value commutative, the maximum eigenvalue commutative) metaplectic transforms of complex-valued functions, in the establishment of the corresponding results for the FIO-I-FV and FIO-II-FV.
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