318 research outputs found
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
Wigner Analysis of Operators. Part II: Schrödinger Equations
No full textWe study the phase-space concentration of the so-called generalized metaplectic operators whose main examples are Schrödinger equations with bounded perturbations. To reach this goal, we perform a so-called -Wigner analysis of the previous equations, as started in Part I, cf. Cordero and Rodino (Appl Comput Harmon Anal 58:85–123, 2022). Namely, the classical Wigner distribution is extended by considering a class of time–frequency representations constructed as images of metaplectic operators acting on symplectic matrices . Sub-classes of these representations, related to covariant symplectic matrices, reveal to be particularly suited for the time–frequency study of the Schrödinger evolution. This testifies the effectiveness of this approach for such equations, highlighted by the development of a related wave front set. We first study the properties of -Wigner representations and related pseudodifferential operators needed for our goal. This approach paves the way to new quantization procedures. As a byproduct, we introduce new quasi-algebras of generalized metaplectic operators containing Schrödinger equations with more general potentials, extending the results contained in the previous works (Cordero et al. in J Math Pures Appl 99(2):219–233, 2013, J Math Phys 55(8):081506, 2014)
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
No full text2002 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
Microlocal analysis for Gelfand--Shilov spaces
No full textWe introduce an anisotropic global wave front set of Gelfand--Shilov
ultradistributions with different indices for regularity and decay at infinity.
The concept is defined by the lack of super-exponential decay along power type
curves in the phase space of the short-time Fourier transform. This wave front
set captures the phase space behaviour of oscillations of power monomial type,
a k a chirp signals. A microlocal result is proved with respect to
pseudodifferential operators with symbol classes that give rise to continuous
operators on Gelfand--Shilov spaces. We determine the wave front set of certain
series of derivatives of the Dirac delta, and exponential functions.Comment: 44 page
- …
