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Elena Cordero

Publications and source records attributed to Elena Cordero.

At least 55 records · Page 3Linked to original sources
arXiv2018

Norm Estimates for $τ$-Pseudodifferential Operators in Wiener Amalgam and Modulation Spaces

We study continuity properties on modulation spaces for $τ$-pseudodifferential operators with symbols $a$ in Wiener amalgam spaces. We obtain boundedness results for $τ\in (0,1)$ whereas, in the end-points $τ=0$ and $τ=1$, the corresponding operators are in general unbounded. Furthermore, for $τ\in (0,1)$, we exhibit a function of $τ$ which is an upper bound for the operator norm. The continuity properties of $τ$-pseudodifferential operators, for any $τ\in [0,1]$, 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 $τ\in [0,1]$, as expected. Key ingredients are uniform continuity estimates for $τ$-Wigner distributions.

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arXiv2017

On Fourier integral operators with Hölder-continuous phase

We study continuity properties in Lebesgue spaces for a class of Fourier integral operators arising in the study of the Boltzmann equation. The phase has a Hölder-type singularity at the origin. We prove boundedness in $L^1$ with a precise loss of decay depending on the Hölder exponent, and we show by counterexamples that a loss occurs even in the case of smooth phases. The results can be seen as a quantitative version of the Beurling-Helson theorem for changes of variables with a Hölder singularity at the origin. The continuity in $L^2$ is studied as well by providing sufficient conditions and relevant counterexamples. The proofs rely on techniques from Time-frequency Analysis.

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arXiv2017

A characterization of modulation spaces by symplectic rotations

This note contains a new characterization of modulation spaces $M^p(\mathbb{R}^n)$, $1\leq p\leq \infty$, by symplectic rotations. Precisely, instead to measure the time-frequency content of a function by using translations and modulations of a fixed window as building blocks, we use translations and metaplectic operators corresponding to symplectic rotations. Technically, this amounts to replace, in the computation of the $M^p(\mathbb{R}^n)$-norm, the integral in the time-frequency plane with an integral on $\mathbb{R}^n\times U(2n,\mathbb{R})$ with respect to a suitable measure, $U(2n,\mathbb{R})$ being the group of symplectic rotations. More conceptually, we are considering a sort of polar coordinates in the time-frequency plane. In this new framework, the Gaussian invariance under symplectic rotations yields to choose Gaussians as suitable window functions. We also provide a similar characterization with the group $U(2n,\mathbb{R})$ being reduced to the $n$-dimensional torus $\mathbb{T}^n$.

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arXiv2017

On the Positivity of Trace Class Operators

The characterization of positivity properties of Weyl operators is a notoriously difficult problem, and not much progress has been made since the pioneering work of Kastler, Loupias, and Miracle-Sole (KLM). In this paper we begin by reviewing and giving simpler proofs of some known results for trace-class Weyl operators; the latter play an essential role in quantum mechanics. We then apply time-frequency analysis techniques to prove a phase space version of the KLM condition; the main tools are Gabor frames and the Wigner formalism. Finally, discrete approximations of the KLM condition, which are tractable numerically, are provided.

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arXiv2017

Kernel theorems for modulation spaces

We deal with kernel theorems for modulation spaces. We completely characterize the continuity of a linear operator on the modulation spaces $M^p$ for every $1\leq p\leq\infty$, by the membership of its kernel to (mixed) modulation spaces. Whereas Feichtinger's kernel theorem (which we recapture as a special case) is the modulation space counterpart of Schwartz' kernel theorem for temperate distributions, our results do not have a couterpart in distribution theory. This reveals the superiority, in some respects, of the modulation space formalism upon distribution theory, as already emphasized in Feichtinger's manifesto for a post-modern harmonic analysis, tailored to the needs of mathematical signal processing. The proof uses in an essential way a discretization of the problem by means of Gabor frames. We also show the equivalence of the operator norm and the modulation space norm of the corresponding kernel. For operators acting on $M^{p,q}$ a similar characterization is not expected, but sufficient conditions for boundedness can be sated in the same spirit.

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arXiv2016

On the symplectic covariance and interferences of time-frequency distributions

We study the covariance property of quadratic time-frequency distributions with respect to the action of the extended symplectic group. We show how covariance is related, and in fact in competition, with the possibility of damping the interferences which arise due to the quadratic nature of the distributions. We also show that the well known fully covariance property of the Wigner distribution in fact characterizes it (up to a constant factor) among the quadratic distributions $L^{2}(\mathbb{R}^{n})\rightarrow C_{0}({ \mathbb{R}^{2n}})$. A similar characterization for the closely related Weyl transform is given as well. The results are illustrated by several numerical experiments for the Wigner and Born-Jordan distributions of the sum of four Gaussian functions in the so-called "diamond configuration".

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arXiv2016

Semi-classical Time-frequency Analysis and Applications

This work represents a first systematic attempt to create a common ground for semi-classical and time-frequency analysis. These two different areas combined together provide interesting outcomes in terms of Schrödinger type equations. Indeed, continuity results of both Schrödinger propagators and their asymptotic solutions are obtained on $\hbar$-dependent Banach spaces, the semi-classical version of the well-known modulation spaces. Moreover, their operator norm is controlled by a constant independent of the Planck's constant $\hbar$. The main tool in our investigation is the joint application of standard approximation techniques from semi-classical analysis and a generalized version of Gabor frames, dependent of the parameter $\hbar$. Continuity properties of more general Fourier integral operators (FIOs) and their sparsity are also investigated.

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arXiv2016

Sharp integral bounds for Wigner distributions

The cross-Wigner distribution $W(f,g)$ of two functions or temperate distributions $f,g$ is a fundamental tool in quantum mechanics and in signal analysis. Usually, in applications in time-frequency analysis $f$ and $g$ belong to some modulation space and it is important to know which modulation spaces $W(f,g)$ belongs to. Although several particular sufficient conditions have been appeared in this connection, the general problem remains open. In the present paper we solve completely this issue by providing the full range of modulation spaces in which the continuity of the cross-Wigner distribution $W(f,g)$ holds, as a function of $f,g$. The case of weighted modulation spaces is also considered. The consequences of our results are manifold: new bounds for the short-time Fourier transform and the ambiguity function, boundedness results for pseudodifferential (in particular, localization) operators and properties of the Cohen class.

math.FA↗
arXiv2016

Sharp results for the Weyl product on modulation spaces

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.

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arXiv2016

Born-Jordan Pseudo-Differential Operators with Symbols in the Shubin Classes

We apply Shubin's theory of global symbol classes $Γ_ρ^{m}$ to the Born-Jordan pseudodifferential calculus we have previously developed. This approach has many conceptual advantages, and makes the relationship between the conflicting Born-Jordan and Weyl quantization methods much more limpid. We give, in particular, precise asymptotic expansions of symbols allowing to pass from Born-Jordan quantization to Weyl quantization, and vice-versa. In addition we state and prove some regularity and global hypoellipticity results.

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arXiv2016

Time-frequency Analysis of Born-Jordan Pseudodifferential Operators

Born-Jordan operators are a class of pseudodifferential operators arising as a generalization of the quantization rule for polynomials on the phase space introduced by Born and Jordan in 1925. The weak definition of such operators involves the Born-Jordan distribution, first introduced by Cohen in 1966 as a member of the Cohen class. We perform a time-frequency analysis of the Cohen kernel of the Born -Jordan distribution, using modulation and Wiener amalgam spaces. We then provide sufficient and necessary conditions for Born-Jordan operators to be bounded on modulation spaces. We use modulation spaces as appropriate symbols classes.

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arXiv2016

On the reduction of the interferences in the Born-Jordan distribution

One of the most popular time-frequency representation is certainly the Wigner distribution. To reduce the interferences coming from its quadratic nature, several related distributions have been proposed, among which the so-called Born-Jordan distribution. It is well known that in the Born-Jordan distribution the ghost frequencies are in fact damped quite well, and the noise is in general reduced. However, the horizontal and vertical directions escape from this general smoothing effect, so that the interferences arranged along these directions are in general kept. Whereas these features are graphically evident on examples and heuristically well understood in the engineering community, there is not at present a mathematical explanation of these phenomena, valid for general signals in L^2 and, more in general, in the space S' of temperate distributions. In the present note we provide such a rigorous study using the notion of wave-front set of a distribution. We use techniques from Time-frequency Analysis, such as the modulation and Wiener amalgam spaces, and also results of microlocal regularity of linear partial differential operators.

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arXiv2015

Gabor analysis for Schrodinger equations and propagation of singularities

We consider the Schrödinger equation \begin{equation*} i \displaystyle\frac{\partial u}{\partial t} +Hu=0,\quad H=a(x,D), \end{equation*} where the Hamiltonian $a(z)$, $z=(x,ξ)$, is assumed real-valued and smooth, with bounded derivatives $|\partial^αa(z)|\leq C_α$, for every $|α|\geq 2$, $z\in\mathbb{R}^{2d}$. For such equation results are known concerning well-posedness of the Cauchy problem for initial data in $L^2(\mathbb{R}^d)$ and local representation of the propagator $e^{it H}$ 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 micro-singularities for $e^{itH}$.

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arXiv2015

On the Invertibility of Born-Jordan Quantization

As a consequence of the Schwartz kernel Theorem, any linear continuous operator $\widehat{A}:$ $\mathcal{S}(\mathbb{R}^{n})\longrightarrow\mathcal{S}^{\prime}(\mathbb{R}^{n})$ can be written in Weyl form in a unique way, namely it is the Weyl quantization of a unique symbol $a\in\mathcal{S}^{\prime}(\mathbb{R}^{2n})$. Hence, dequantization can always be performed, and in a unique way. Despite the importance of this topic in Quantum Mechanics and Time-frequency Analysis, the same issue for the Born-Jordan quantization seems simply unexplored, except for the case of polynomial symbols, which we also review in detail. In this paper we show that any operator $\widehat{A}$ as above can be written in Born-Jordan form, although the representation is never unique if one allows general temperate distributions as symbols. Then we consider the same problem when the space of temperate distributions is replaced by the space of smooth slowly increasing functions which extend to entire function in $\mathbb{C}^{2n}$, with a growth at most exponential in the imaginary directions. We prove again the validity of such a representation, and we determine a sharp threshold for the exponential growth under which the representation is unique. We employ techniques from the theory of division of distributions.

math.FA↗
arXiv2015

Gabor representations of evolution operators

We perform a time-frequency analysis of Fourier multipliers and, more generally, pseudodifferential operators with symbols of Gevrey, analytic and ultra-analytic regularity. As an application we show that Gabor frames, which provide optimally sparse decompositions for Schroedinger-type propagators, reveal to be an even more efficient tool for representing solutions to a wide class of evolution operators with constant coefficients, including weakly hyperbolic and parabolic-type operators. Besides the class of operators, the main novelty of the paper is the proof of super-exponential (as opposite to super-polynomial) off-diagonal decay for the Gabor matrix representation.

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arXiv2015

Exponentially sparse representations of Fourier integral operators

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.

math.FA↗
arXiv2015

Wave packet analysis of Schrodinger equations in analytic function spaces

We consider a class of linear Schroedinger 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.

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