1,721,186 research outputs found
Remarks on lower bounds for pseudo-differential operators
No full textAbstractWe study lower bounds for pseudo-differential operators with multiple characteristics. The principal symbol is assumed positive, vanishing exactly to the order k⩾2 on a smooth manifold Σ. Under an additional positivity assumption on the Jth Taylor polynomial of the sub-principal symbol at Σ, 0⩽J⩽k/2−1, using the Fefferman–Phong inequality we get a lower bound with gain of k/(k−J−1) derivatives
SG-pseudo-differential operators and weak hyperbolicity
Get PDF2002 Mathematics Subject Classification: 35S05, 47G30, 58J42.We consider a class of pseudo-differential operators globally defined in R^n. For them we discuss trace functionals, distribution of eigenvalues, essential spectrum and weak hyperbolicity.*The authors are supported by NATO, PST.CLG.979347, Collaborative Linkage Grant CNR- BAN and FIRB 2001, COFIN 2002, Italy
Inhomogeneous Gevrey classes and ultradistributions
No full textThe inhomogeneous Gevrey classes, defined in terms of Fourier transform, are a natural extension of the standard Gevrey classes. We find equivalent characterizations and discuss algebraic and topological properties. We therefore introduce the dual spaces, the inhomogeneous ultradistributions, giving some equivalent definitions and corresponding algebraic and topological properties; in particular, a version of the Paley–Wiener–Schwartz theorem is proved in our framework. Finally, as an important example, the multianisotropic Gevrey classes and ultradistributions are considered
Sparsity of Gabor representation of Schrodinger propagators
No full textAbstractRecent papers show how tight frames of curvelets and shearlets provide optimally sparse representation of hyperbolic-type Fourier integral operators (FIOs) [E.J. Candés, L. Demanet, Curvelets and Fourier integral operators, C. R. Math. Acad. Sci. Paris 336 (5) (2003) 395–398; E.J. Candés, L. Demanet, The curvelet representation of wave propagators is optimally sparse, Comm. Pure Appl. Math. 58 (2005) 1472–1528; E.J. Candés, L. Demanet, L. Ying, Fast computation of Fourier integral operators, SIAM J. Sci. Comput. 29 (6) (2007) 2464–2493; K. Guo, D. Labate, Sparse shearlet representation of Fourier integral operators, Electron. Res. Announc. Math. Sci. 14 (2007) 7–19]. 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 [B. Helffer, Théorie spectrale pour des operateurs globalement elliptiques, Astérisque, Société Mathématique de France, 1984; B. Helffer, D. Robert, Comportement asymptotique precise du spectre d'operateurs globalement elliptiques dans Rd, Sem. Goulaouic–Meyer–Schwartz 1980–81, École Polytechnique, 1980, Exposé II], and hence studied by many other authors, see, e.g., [A. Boulkhemair, Remarks on a Wiener type pseudodifferential algebra and Fourier integral operators, Math. Res. Lett. 4 (1997) 53–67; F. Concetti, J. Toft, Schatten–von Neumann properties for Fourier integral operators with non-smooth symbols I, Ark. Mat., in press]. 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
A Unified Approach to Time–Frequency Representations and Generalized Spectrograms
No full textTo overcome the impossibility of representing the energy of a signal simultaneously in time and frequency, many time–frequency representations have been introduced in the literature. Some of these are recalled in the Introduction. In this work, we propose a unified approach to the previous theory by means of metaplectic Wigner distributions WA, with A a symplectic matrix in Sp(2d,R), which were introduced by Cordero and Rodino (Appl Comput Harmon Anal 58:85–123, 2022) and then widely studied in subsequent papers. Namely, the short-time Fourier transform and the most popular members of Cohen’s class can be represented via metaplectic Wigner distributions. In particular, we introduce metaplectic spectrograms, which contain the classical ones and their variations arising from the tau-Wigner distributions of Boggiatto et al. (Trans Am Math Soc 362(9):4955–4981, 2010). We provide a complete characterization of those A-Wigner distributions which give rise to generalized spectrograms. This characterization is related to the block decomposition of the symplectic matrix A. Moreover, a characterization of the boundedness of both A-Wigner distributions and related metaplectic pseudodifferential operators is provided
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