arXiv subjects

Elena Cordero

Publications and source records attributed to Elena Cordero.

At least 37 records · Page 2Linked to original sources

arXiv2023

Characterization of modulation spaces by symplectic representations and applications to Schrödinger equations

In the last twenty years modulation spaces, introduced by H. G. Feichtinger in 1983, have been successfully addressed to the study of signal analysis, PDE's, pseudodifferential operators, quantum mechanics, by hundreds of contributions. In 2011 M. de Gosson showed that the time-frequency representation Short-time Fourier Transform (STFT), which is the tool to define modulation spaces, can be replaced by the Wigner distribution. This idea was further generalized to $τ$-Wigner representations in [9]. In this paper time-frequency representations are viewed as images of symplectic matrices via metaplectic operators. This new perspective highlights that the protagonists of time-frequency analysis are metaplectic operators and symplectic matrices $\mathcal{A} \in Sp(2d,\mathbb{R})$. We find conditions on $\mathcal{A}$ for which the related symplectic time-frequency representation $W_\mathcal{A}$ can replace the STFT and give equivalent norms for weighted modulation spaces. In particular, we study the case of covariant matrices $\mathcal{A}$, i.e., their corresponding $W_\mathcal{A}$ are members of the Cohen class. Finally, we show that symplectic time-frequency representations $W_\mathcal{A}$ can be efficiently employed in the study of Schrödinger equations. This new approach may have further applications in quantum mechanics and PDE's.

math.FA↗

arXiv2023

Symplectic Analysis of Time-Frequency Spaces

We present a different symplectic point of view in the definition of weighted modulation spaces $M^{p,q}_m(\mathbb{R}^d)$ and weighted Wiener amalgam spaces $W(\mathcal{F} L^p_{m_1},L^q_{m_2})(\mathbb{R}^d)$. All of the classical time-frequency representations, such as the short-time Fourier transform (STFT), the $τ$-Wigner distributions and the ambiguity function, can be written as metaplectic Wigner distributions $μ(\mathcal{A})(f\otimes \bar{g})$, where $μ(\mathcal{A})$ is the metaplectic operator and $\mathcal{A}$ is the associated symplectic matrix. Namely, time-frequency representations can be represented as images of metaplectic operators, which become the real protagonists of time-frequency analysis. In [E. Cordero and L. Rodino (2022) "Characterization of Modulation Spaces by symplectic representations and applications to Schrödinger equations", arXiv:2204.14124], the authors suggest that any metaplectic Wigner distribution that satisfies the so-called "shift-invertibility" condition can replace the STFT in the definition of modulation spaces. In this work, we prove that shift-invertibility alone is not sufficient, but it has to be complemented by an upper-triangularity condition for this characterization to hold, whereas a lower-triangularity property comes in to play for Wiener amalgam spaces. The shift-invertibility property is necessary: Ryhaczek and and conjugate Ryhaczek distributions are not shift-invertible and they fail the characterization of the above spaces. We also exhibit examples of shift-invertible distributions without upper-tryangularity condition which do not define modulation spaces. Finally, we provide new families of time-frequency representations that characterize modulation spaces, with the purpose of replacing the time-frequency shifts with other atoms that allow to decompose signals differently, with possible new outcomes in applications.

math.AP↗

arXiv2022

Quasi-Banach algebras and Wiener properties for pseudodifferential and generalized metaplectic operators

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 [11], 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. [7].

math.FA↗

arXiv2022

Wigner Analysis of Operators. Part II: Schrödinger equations

We 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 $\mathcal{A}$-Wigner analysis of the previous equations, as started in Part I, cf. [14]. 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 $\mathcal{A}\in Sp(2d,\mathbb{R})$. 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 $\mathcal{A}$-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 [8,9].

math.AP↗

arXiv2022

Quasi-Banach modulation spaces and localization operators on locally compact abelian groups

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 quasi-lattices, 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.

math.FA↗

arXiv2022

Comparisons between Fourier and STFT multipliers: the smoothing effect of the Short-time Fourier Transform

We study the connection between STFT multipliers $A^{g_1,g_2}_{1\otimes m}$ having windows $g_1,g_2$, symbols $a(x,ω)=(1\otimes m)(x,ω)=m(ω)$, $(x,ω)\in\mathbb{R}^{2d}$, and the Fourier multipliers $T_{m_2}$ with symbol $m_2$ on $\mathbb{R}^d$. We find sufficient and necessary conditions on symbols $m,m_2$ and windows $g_1,g_2$ for the equality $T_{m_2}= A^{g_1,g_2}_{1\otimes m}$. For $m=m_2$ the former equality holds only for particular choices of window functions in modulation spaces, whereas it never occurs in the realm of Lebesgue spaces. In general, the STFT multiplier $A^{g_1,g_2}_{1\otimes m}$, also called localization operator, presents a smoothing effect due to the so-called two-window short-time Fourier transform which enters in the definition of $A^{g_1,g_2}_{1\otimes m}$. As a by-product we prove necessary conditions for the continuity of anti-Wick operators $A^{g,g}_{1\otimes m}: L^p\to L^q$ having multiplier $m$ in weak $L^r$ spaces. Finally, we exhibit the related results for their discrete counterpart: in this setting STFT multipliers are called Gabor multipliers whereas Fourier multiplier are better known as linear time invariant (LTI) filters.

math.FA↗

arXiv2021

Characterization of smooth symbol classes by Gabor matrix decay

For $m\in\mathbb{R}$ we introduce the symbol classes $S^m$, $m\in\mathbb{R}$, consisting of smooth functions $σ$ on $\mathbb{R}^{2d}$ such that $|\partial^ασ(z)|\leq C_α(1+|z|^2)^{m/2}$, $z\in\mathbb{R}^{2d}$, and we show that can be characterized by an intersection of different types of modulation spaces. In the case $m=0$ we recapture the Hörmander class $S^0_{0,0}$ that can be obtained by intersection of suitable Besov spaces as well. Such spaces contain the Shubin classes $Γ^m_ρ$, $0<ρ\leq1$, and can be viewed as their limit case $ρ=0$. We exhibit almost diagonalization properties for the Gabor matrix of $τ$-pseudodifferential operators with symbols in such classes, extending the characterization proved by Gröchenig and Rzeszotnik. Finally, we compute the Gabor matrix of a Born-Jordan operator, which allows to prove new boundedness results for such operators.

math.FA↗

arXiv2021

Wigner analysis of operators. Part I: pseudodifferential operators and wave fronts

We perform Wigner analysis of linear operators. Namely, the standard time-frequency representation \emph{Short-time Fourier Transform} (STFT) is replaced by the $\mathcal{A}$-\emph{Wigner distribution} defined by $W_{\mathcal A} (f)=μ({\mathcal A})(f\otimes\bar{f})$, where ${\mathcal A}$ is a $4d\times 4d$ symplectic matrix and $μ({\mathcal A})$ is an associate metaplectic operator. Basic examples are given by the so-called $τ$-Wigner distributions. Such representations provide a new characterization for modulation spaces when $τ\in (0,1)$. Furthermore, they can be efficiently employed in the study of the off-diagonal decay for pseudodifferential operators with symbols in the Sjöstrand class (in particular, in the Hörmander class $S^0_{0,0}$). The novelty relies on defining time-frequency representations via metaplectic operators, developing a conceptual framework and paving the way for a new understanding of quantization procedures. We deduce micro-local properties for pseudodifferential operators in terms of the Wigner wave front set. Finally, we compare the Wigner with the global Hörmander wave front set and identify the possible presence of a ghost region in the Wigner wave front. \par In the second part of the paper applications to Fourier integral operators and Schrödinger equations will be given.

math.AP↗

arXiv2020

On the local well-posedness of the nonlinear heat equation associated to the fractional Hermite operator in modulation spaces

In this note we consider the nonlinear heat equation associated to the fractional Hermite operator $H^β=(-Δ+|x|^2)^β$, $0<β\leq 1$. 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 $S^m_{0,0}$, $m\in\mathcal{R}$.

math.AP↗

arXiv2020

Decay and Smoothness for Eigenfunctions of Localization Operators

We study decay and smoothness properties for eigenfunctions of compact localization operators. Operators with symbols a in the wide modulation space M^{p,\infty} (containing the Lebesgue space L^p), p<\infty, and windows \f_1,\f_2 in the Schwartz class are known to be compact. We show that their L^2-eigenfuctions with non-zero eigenvalues are indeed highly compressed onto a few Gabor atoms. Similarly, for symbols a in the weighted modulation spaces M^{\infty}_{v_s\otimes 1} (\rdd), s>0 (subspaces of M^{p,\infty}(\rdd), p>2d/s) the L^2-eigenfunctions of the localization operator are actually Schwartz functions. An important role is played by quasi-Banach Wiener amalgam and modulation spaces. As a tool, new convolution relations for modulation spaces and multiplication relations for Wiener amalgam spaces in the quasi-Banach setting are exhibited.

math.FA↗

arXiv2020

Dispersion, spreading and sparsity of Gabor wave packets for metaplectic and Schrödinger operators

Sparsity properties for phase-space representations of several types of operators have been extensively studied in recent papers, including pseudodifferential, Fourier integral and metaplectic operators, with applications to time-frequency analysis of Schrödinger-type evolution equations. It has been proved that such operators are approximately diagonalized by Gabor wave packets. While the latter are expected to undergo some spreading phenomenon, there is no record of this issue in the aforementioned results. In this paper we prove refined estimates for the Gabor matrix of metaplectic operators, also of generalized type, where sparsity, spreading and dispersive properties are all noticeable. We provide applications to the propagation of singularities for the Schrödinger equation.

math.FA↗

arXiv2020

Note on the Wigner distribution and Localization Operators in the quasi-Banach setting

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 $S_p$, $0<p<1$, are exhibited. As a byproduct, sharp continuity results for the Wigner distribution are also presented.

math.FA↗

arXiv2019

Signal Analysis using Born-Jordan-type Distribution

In this note we exhibit recent advances in signal analysis via time-frequency distributions. New members of the Cohen class, generalizing the Wigner distribution, reveal to be effective in damping artefacts of some signals. We will survey their main properties and drawbacks and present open problems related to such phenomena.

math.FA↗

arXiv2019

Linear perturbations of the Wigner transform and the Weyl quantization

We study a class of quadratic time-frequency representations that, roughly speaking, are obtained by linear perturbations of the Wigner transform. They satisfy Moyal's formula by default and share many other properties with the Wigner transform, but in general they do not belong to Cohen's class. We provide a characterization of the intersection of the two classes. To any such time-frequency representation, we associate a pseudodifferential calculus. We investigate the related quantization procedure, study the properties of the pseudodifferential operators, and compare the formalism with that of the Weyl calculus.

math.FA↗

arXiv2019

Linear perturbations of the Wigner distribution and the Cohen's class

The Wigner distribution is a milestone of Time-frequency Analysis. In order to cope with its drawbacks while preserving the desirable features that made it so popular, several kind of modifications have been proposed. This contributions fits into this perspective. We introduce a family of phase-space representations of Wigner type associated with invertible matrices and explore their general properties. As main result, we provide a characterization for the Cohen's class. This feature suggests to interpret this family of representations as linear perturbations of the Wigner distribution. We show which of its properties survive under linear perturbations and which ones are truly distinctive of its central role.

math.FA↗

arXiv2018

Generalized Born--Jordan Distributions and Applications

The quadratic nature of the Wigner distribution causes the appearance of unwanted interferences. This is the reason why engineers, mathematicians and physicists look for related time-frequency distributions, many of them are members of the Cohen class. Among them, the Born-Jordan distribution has recently attracted the attention of many authors, since the so-called ghost frequencies are damped quite well, and the noise is in general reduced. The very insight relies on the kernel of such a distribution, which contains the sinus cardinalis (sinc), which can be viewed as the Fourier transform, of the first B-Spline. Replacing the function B-Spline with the spline or order n, on the Fourier side we obtain the n-th power of sinc, whose decay at infinity increases with n. We introduce the corresponding Cohen's Kernel and study the properties of the related time-frequency distribution which we call generalized Born--Jordan distribution.

math.FA↗

arXiv2018

Almost diagonalization of $τ$-pseudodifferential operators with symbols in Wiener amalgam and modulation spaces

In this paper we focus on the almost-diagonalization properties of $τ$-pseudodifferential operators using techniques from time-frequency analysis. Our function spaces are modulation spaces and the special class of Wiener amalgam spaces arising by considering the action of the Fourier transform of modulation spaces. A particular example is provided by the Sjöstrand class, for which Gröchenig exhibited the almost diagonalization of Weyl operators. We shall show that such result can be extended to any $τ$-pseudodifferential operator, for $τ\in [0,1]$, also with symbol in weighted Wiener amalgam spaces. As a consequence, we infer boundedness, algebra and Wiener properties for $τ$-pseudodifferential operators on Wiener amalgam and modulation spaces.

math.FA↗

arXiv2018

Strichartz Estimates for the Schrödinger Equation

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 well-posedness.

math.AP↗