Applied and Computational Harmonic Analysis, Jul 1, 2004
The theory of localized frames is refined to include quasi-Banach spaces and spaces with multiple... more The theory of localized frames is refined to include quasi-Banach spaces and spaces with multiple generators. Applications are given to nonlinear approximation with frames and to the convergence of the iterative frame algorithm in finer norms, and to the characterization of Besov spaces with wavelet frames.
We give sufficient and necessary conditions on the Lebesgue exponents for the Weyl product to be ... more We give sufficient and necessary conditions on the Lebesgue exponents for the Weyl product to be bounded on modulation spaces. The sufficient conditions are obtained as the restriction to N = 2 of a result valid for the N -fold Weyl product. As a byproduct, we obtain sharp conditions for the twisted convolution to be bounded on Wiener amalgam spaces.
We consider Schrödinger equations with real-valued smooth Hamiltonians, and non-smooth bounded ps... more We consider Schrödinger equations with real-valued smooth Hamiltonians, and non-smooth bounded pseudo-differential potentials, whose symbols may be not even 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.
Journal of Mathematical Analysis and Applications, Mar 1, 2019
We study continuity properties on modulation spaces for τ -pseudodifferential operators Op τ (a) ... more We study continuity properties on modulation spaces for τ -pseudodifferential operators Op τ (a) with symbols a in Wiener amalgam spaces. We obtain boundedness results for τ ∈ (0, 1) whereas, in the end-points τ = 0 and τ = 1, the corresponding operators are in general unbounded. Furthermore, for τ ∈ (0, 1), we exhibit a function of τ which is an upper bound for the operator norm. The continuity properties of Op τ (a), for any τ ∈ [0, 1], with symbols a in modulation spaces are well known. Here we find an upper bound for the operator norm which does not depend on the parameter τ ∈ [0, 1], as expected. Key ingredients are uniform continuity estimates for τ -Wigner distributions. R d R d e 2πi(x-y)ξ a((1 -τ )x + τ y, ξ)f (y) dydξ, f ∈ S(R d ).
We consider a class of linear Schrödinger equations in R d , with analytic symbols. We prove a gl... more We consider a class of linear Schrödinger equations in R d , with analytic symbols. We prove a global-in-time integral representation for the corresponding propagator as a generalized Gabor multiplier with a window analytic and decaying exponentially at infinity, which is transported by the Hamiltonian flow. We then provide three applications of the above result: the exponential sparsity in phase space of the corresponding propagator with respect to Gabor wave packets, a wave packet characterization of Fourier integral operators with analytic phases and symbols, and the propagation of analytic singularities.
We first give a short survey on the methods of the Microlocal Analysis; in particular we recall s... more We first give a short survey on the methods of the Microlocal Analysis; in particular we recall some basic facts concerning the theory of the pseudo-differential operators. We then present two applications, namely: we discuss lower bounds for operators with multiple characteristics and we give a new formula of composition for Wick operators.
We generalize the results for Banach algebras of pseudodifferential operators obtained by Gröchen... more We generalize the results for Banach algebras of pseudodifferential operators obtained by Gröchenig and Rzeszotnik in [24] to quasi-algebras of Fourier integral operators. Namely, we introduce quasi-Banach algebras of symbol classes for Fourier integral operators that we call generalized metaplectic operators, including pseudodifferential operators. This terminology stems from the pioneering work on Wiener algebras of Fourier integral operators , which we generalize to our framework. This theory finds applications in the study of evolution equations such as the Cauchy problem for the Schrödinger equation with bounded perturbations, cf. .
Applied and Computational Harmonic Analysis, May 1, 2009
Recent papers show how tight frames of curvelets and shearlets provide optimally sparse represent... more Recent papers show how tight frames of curvelets and shearlets provide optimally sparse representation of hyperbolic-type Fourier Integral Operators (FIOs) . In this paper we address to another class of FIOs, employed by Helffer and Robert to study spectral properties of globally elliptic operators of Quantum Mechanics , and hence studied by many other authors, see, e.g., . An example is provided by the resolvent of the Cauchy problem for the Schrödinger equation with a quadratic Hamiltonian. We show that Gabor frames provide optimally sparse representations of such operators. Numerical examples for the Schrödinger case demonstrate the fast computation of these operators.
Applied and Computational Harmonic Analysis, May 1, 2004
The theory of localized frames is refined to include quasi-Banach spaces and spaces with multiple... more The theory of localized frames is refined to include quasi-Banach spaces and spaces with multiple generators. Applications are given to nonlinear approximation with frames and to the convergence of the iterative frame algorithm in finer norms, and to the characterization of Besov spaces with wavelet frames.
Time-frequency analysis have played a crucial role in the development of localization operators i... more Time-frequency analysis have played a crucial role in the development of localization operators in the last twenty years. We present its applications to the study of boundedness and Schatten Class property for such operators. In particular, new sufficient conditions for such operators to belong to the Schatten-von Neumann Class S p (L 2 (R d )), 0 < p < 1, are exhibited. As a byproduct, sharp continuity results for the Wigner distribution are also presented.
We give sufficient and necessary conditions on the Lebesgue exponents for the Weyl product to be ... more We give sufficient and necessary conditions on the Lebesgue exponents for the Weyl product to be bounded on modulation spaces. The sufficient conditions are obtained as the restriction to N = 2 of a result valid for the N -fold Weyl product. As a byproduct, we obtain sharp conditions for the twisted convolution to be bounded on Wiener amalgam spaces.
The objective of this paper is to report on recent progress on Strichartz estimates for the Schrö... more The objective of this paper is to report on recent progress on Strichartz estimates for the Schrödinger equation and to present the state-of-the-art. These estimates have been obtained in Lebesgue spaces, Sobolev spaces and, recently, in Wiener amalgam and modulation spaces. We present and compare the different technicalities. Then, we illustrate applications to wellposedness.
In this note we consider the nonlinear heat equation associated to the fractional Hermite operato... more In this note we consider the nonlinear heat equation associated to the fractional Hermite operator H β = (-∆+|x| 2 ) β , 0 < β ≤ 1. We show the local solvability of the related Cauchy problem in the framework of modulation spaces. The result is obtained by combining tools from microlocal and time-frequency analysis. As a byproduct, we compute the Gabor matrix of pseudodifferential operators with symbols in the Hörmander class S m 0,0 , m ∈ R.
Sampling theory, signal processing, and data analysis, Mar 25, 2024
We provide a comprehensive overview of the theoretical framework surrounding modulation spaces an... more We provide a comprehensive overview of the theoretical framework surrounding modulation spaces and their characterizations, particularly focusing on the role of metaplectic operators and time-frequency representations. We highlight the metaplectic action which is hidden in their construction and guarantees equivalent (quasi-)norms for such spaces. In particular, this work provides new characterizations via the submanifold of shift-invertible symplectic matrices. Similar results hold for the Wiener amalgam spaces. Frames • Time-frequency analysis • Modulation spaces • Wiener amalgam spaces • Time-frequency representations • Metaplectic group • Symplectic group Mathematics Subject Classification 42C15 • 42B35 • 42A38 Communicated by Martin Ehler.
It is well known that the matrix of a metaplectic operator with respect to phase-space shifts is ... more It is well known that the matrix of a metaplectic operator with respect to phase-space shifts is concentrated along the graph of a linear symplectic map. We show that the algebra generated by metaplectic operators and by pseudodifferential operators in a Sjöstrand class enjoys the same decay properties. We study the behavior of these generalized metaplectic operators and represent them by Fourier integral operators. Our main result shows that the one-parameter group generated by a Hamiltonian operator with a potential in the Sjöstrand class consists of generalized metaplectic operators. As a consequence, the Schrödinger equation preserves the phase-space concentration, as measured by modulation space norms.
A general principle says that the matrix of a Fourier integral operator with respect to wave pack... more A general principle says that the matrix of a Fourier integral operator with respect to wave packets is concentrated near the curve of propagation. We prove a precise version of this principle for Fourier integral operators with a smooth phase and a symbol in the Sjöstrand class and use Gabor frames as wave packets. The almost diagonalization of such Fourier integral operators suggests a specific approximation by (a sum of) elementary operators, namely modified Gabor multipliers. We derive error estimates for such approximations. The methods are taken from time-frequency analysis.
We investigate the sparsity of the Gabor-matrix representation of Fourier integral operators with... more We investigate the sparsity of the Gabor-matrix representation of Fourier integral operators with a phase having quadratic growth. It is known that such an infinite matrix is sparse and well organized, being in fact concentrated along the graph of the corresponding canonical transformation. Here we show that, if the phase and symbol have a regularity of Gevrey type of order s > 1 or analytic (s = 1), the above decay is in fact sub-exponential or exponential, respectively. We also show by a counterexample that ultra-analytic regularity (s < 1) does not give super-exponential decay. This is in sharp contrast to the more favorable case of pseudodifferential operators, or even (generalized) metaplectic operators, which are treated as well.
We introduce new quasi-Banach modulation spaces on locally compact abelian (LCA) groups which coi... more We introduce new quasi-Banach modulation spaces on locally compact abelian (LCA) groups which coincide with the classical ones in the Banach setting and prove their main properties. Then we study Gabor frames on quasilattices, significantly extending the original theory introduced by Gröchenig and Strohmer. These issues are the key tools in showing boundedness results for Kohn-Nirenberg and localization operators on modulation spaces and studying their eigenfunctions' properties. In particular, the results in the Euclidean space are recaptured.
We consider the Schrödinger equation i ∂u ∂t + Hu = 0, H = a(x, D), where the Hamiltonian a(z), z... more We consider the Schrödinger equation i ∂u ∂t + Hu = 0, H = a(x, D), where the Hamiltonian a(z), z = (x, η), is assumed real-valued and smooth, with bounded derivatives |∂ α a(z)| ≤ C α , for every |α| ≥ 2, z ∈ R 2d . For such equation results are known concerning well-posedness of the Cauchy problem for initial data in L 2 (R d ) and local representation of the propagator e itH by means of Fourier integral operators. In the present paper we give a global expression for e itH in terms of Gabor analysis and we deduce boundedness in modulation spaces. Moreover, by using time-frequency techniques, we obtain a result of propagation of microsingularities for e itH .
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