arXiv subjects
Publications and source records attributed to Fabio Nicola.
We establish some fixed-time decay estimates in Lebesgue spaces for the fractional heat propagator $e^{-tH^β}$, $t, β>0$, associated with the harmonic oscillator $H=-Δ+ |x|^2$. We then prove some local and global wellposedness results for nonlinear fractional heat equations.
We consider the problem of the maximum concentration in a fixed measurable subset $Ω\subset\mathbb{R}^{2d}$ of the time-frequency space for functions $f\in L^2(\mathbb{R}^{d})$. The notion of concentration can be made mathematically precise by considering the $L^p$-norm on $Ω$ of some time-frequency distribution of $f$ such as the ambiguity function $A(f)$. We provide a positive answer to an open maximization problem, by showing that for every subset $Ω\subset\mathbb{R}^{2d}$ of finite measure and every $1\leq p<\infty$, there exists an optimizer for \[ \sup\{\|A(f)\|_{L^p(Ω)}:\ f\in L^2(\mathbb{R}^{d}),\ \|f\|_{L^2}=1 \}. \] The lack of weak upper semicontinuity and the invariance under time-frequency shifts make the problem challenging. The proof is based on concentration compactness with time-frequency shifts as dislocations, and certain integral bounds and asymptotic decoupling estimates for the ambiguity function. We also discuss the case $p=\infty$ and related optimization problems for the time correlation function, the cross-ambiguity function with a fixed window, and for functions in the modulation spaces $M^q(\mathbb{R}^{d})$, $0<q<2$, equipped with continuous or discrete-type (quasi-)norms.
Within the mathematical analysis of deep convolutional neural networks, the wavelet scattering transform introduced by Stéphane Mallat is a unique example of how the ideas of multiscale analysis can be combined with a cascade of modulus nonlinearities to build a nonexpansive, translation invariant signal representation with provable geometric stability properties, namely Lipschitz continuity to the action of small $C^2$ diffeomorphisms - a remarkable result for both theoretical and practical purposes, inherently depending on the choice of the filters and their arrangement into a hierarchical architecture. In this note, we further investigate the intimate relationship between the scattering structure and the regularity of the deformation in the Hölder regularity scale $C^α$, $α>0$. We are able to precisely identify the stability threshold, proving that stability is still achievable for deformations of class $C^α$, $α>1$, whereas instability phenomena can occur at lower regularity levels modelled by $C^α$, $0\le α<1$. While the behaviour at the threshold given by Lipschitz (or even $C^1$) regularity remains beyond reach, we are able to prove a stability bound in that case, up to $\varepsilon$ losses.
A well-known version of the uncertainty principle on the cyclic group $\mathbb{Z}_N$ states that for any couple of functions $f,g\in\ell^2(\mathbb{Z}_N)\setminus\{0\}$, the short-time Fourier transform $V_g f$ has support of cardinality at least $N$. This result can be regarded as a time-frequency version of the celebrated Donoho-Stark uncertainty principle on $\mathbb{Z}_N$. Unlike the Donoho-Stark principle, however, a complete identification of the extremals is still missing. In this note we provide an answer to this problem by proving that the support of $V_g f$ has cardinality $N$ if and only if it is a coset of a subgroup of order $N$ of $\mathbb{Z}_N\times \mathbb{Z}_N$. Also, we completely identify the corresponding extremal functions $f,g$. Besides translations and modulations, the symmetries of the problem are encoded by certain metaplectic operators associated with elements of ${\rm SL}(2,\mathbb{Z}_{N/a})$, where $a$ is a divisor of $N$. Partial generalizations are given to finite Abelian groups.
In this paper we solve an open problem concerning the characterization of those measurable sets $Ω\subset \mathbb{R}^{2d}$ that, among all sets having a prescribed Lebesgue measure, can trap the largest possible energy fraction in time-frequency space, where the energy density of a generic function $f\in L^2(\mathbb{R}^d)$ is defined in terms of its Short-time Fourier transform (STFT) $\mathcal{V} f(x,ω)$, with Gaussian window. More precisely, given a measurable set $Ω\subset\mathbb{R}^{2d}$ having measure $s> 0$, we prove that the quantity \[ Φ_Ω=\max\Big\{\int_Ω|\mathcal{V} f(x,ω)|^2\,dxdω: f\in L^2(\mathbb{R}^d),\ \|f\|_{L^2}=1\Big\}, \] is largest possible if and only if $Ω$ is equivalent, up to a negligible set, to a ball of measure $s$, and in this case we characterize all functions $f$ that achieve equality. This result leads to a sharp uncertainty principle for the "essential support" of the STFT (when $d=1$, this can be summarized by the optimal bound $Φ_Ω\leq 1-e^{-|Ω|}$, with equality if and only if $Ω$ is a ball). Our approach, using techniques from measure theory after suitably rephrasing the problem in the Fock space, also leads to a local version of Lieb's uncertainty inequality for the STFT in $L^p$ when $p\in [2,\infty)$, as well as to $L^p$-concentration estimates when $p\in [1,\infty)$, thus proving a related conjecture. In all cases we identify the corresponding extremals.
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.
We study the Hermite operator $H=-Δ+|x|^2$ in $\mathbb{R}^d$ and its fractional powers $H^β$, $β>0$ in phase space. Namely, we represent functions $f$ via the so-called short-time Fourier, alias Fourier-Wigner or Bargmann transform $V_g f$ ($g$ being a fixed window function), and we measure their regularity and decay by means of mixed Lebesgue norms in phase space of $V_g f$, that is in terms of membership to modulation spaces $M^{p,q}$, $0< p,q\leq \infty$. We prove the complete range of fixed-time estimates for the semigroup $e^{-tH^β}$ when acting on $M^{p,q}$, for every $0< p,q\leq \infty$, exhibiting the optimal global-in-time decay as well as phase-space smoothing. As an application, we establish global well-posedness for the nonlinear heat equation for $H^β$ with power-type nonlinearity (focusing or defocusing), with small initial data in modulation spaces or in Wiener amalgam spaces. We show that such a global solution exhibits the same optimal decay $e^{-c t}$ as the solution of the corresponding linear equation, where $c=d^β$ is the bottom of the spectrum of $H^β$. This is in sharp contrast to what happens for the nonlinear focusing heat equation without potential, where blow-up in finite time always occurs for (even small) constant initial data - hence in $M^{\infty,1}$.
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.
In the first part of the paper we provide a survey of recent results concerning the problem of pointwise convergence of integral kernels in Feynman path integral, obtained by means of time-frequency analysis techniques. We then focus on exceptional times, where the previous results do not hold, and we show that weaker forms of convergence still occur. In conclusion we offer some clues about possible physical interpretation of exceptional times.
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.
In this note we provide a negative answer to a question raised by M. Kreisel concerning a condition on the short-time Fourier transform that would imply the HRT conjecture. In particular we provide a new type of uncertainty principle for the short-time Fourier transform which forbids the arrangement of an arbitrary "bump with fat tail" profile.
Estimates for matrix coefficients of unitary representations of semisimple Lie groups have been studied for a long time, starting with the seminal work by Bargmann, by Ehrenpreis and Mautner, and by Kunze and Stein. Two types of estimates have been established: on the one hand, $L^p$ estimates, which are a dual formulation of the Kunze--Stein phenomenon, and which hold for all matrix coefficients, and on the other pointwise estimates related to asymptotic expansions at infinity, which are more precise but only hold for a restricted class of matrix coefficients. In this paper we prove a new type of estimate for the irreducibile unitary representations of $\mathrm{SL}(2,\mathbb{R})$ and for the so-called metaplectic representation, which we believe has the best features of, and implies, both forms of estimate described above. As an application outside representation theory, we prove a new $L^2$ estimate of dispersive type for the free Schrödinger equation in $\mathbb{R}^n$.
We study path integrals in the Trotter-type form for the Schrödinger equation, where the Hamiltonian is the Weyl quantization of a real-valued quadratic form perturbed by a potential $V$ in a class encompassing that - considered by Albeverio and Itô in celebrated papers - of Fourier transforms of complex measures. Essentially, $V$ is bounded and has the regularity of a function whose Fourier transform is in $L^1$. Whereas the strong convergence in $L^2$ in the Trotter formula, as well as several related issues at the operator norm level are well understood, the original Feynman's idea concerned the subtler and widely open problem of the pointwise convergence of the corresponding probability amplitudes, that are the integral kernels of the approximation operators. We prove that, for the above class of potentials, such a convergence at the level of the integral kernels in fact occurs, uniformly on compact subsets and for every fixed time, except for certain exceptional time values for which the kernels are in general just distributions. Actually, theorems are stated for potentials in several function spaces arising in Harmonic Analysis, with corresponding convergence results. Proofs rely on Banach algebras techniques for pseudo-differential operators acting on such function spaces.
We study the convergence in $L^2$ of the time slicing approximation of Feynman path integrals under low regularity assumptions on the potential. Inspired by the custom in Physics and Chemistry, the approximate propagators considered here arise from a series expansion of the action. The results are ultimately based on function spaces, tools and strategies which are typical of Harmonic and Time-frequency analysis.
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.
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.
We study the continuity on the modulation spaces $M^{p,q}$ of Fourier multipliers with symbols of the type $e^{iμ(ξ)}$, for some real-valued function $μ(ξ)$. A number of results are known, assuming that the derivatives of order $\geq 2$ of the phase $μ(ξ)$ are bounded or, more generally, that its second derivatives belong to the Sjöstrand class $M^{\infty,1}$. Here we extend those results, by assuming that the second derivatives lie in the bigger Wiener amalgam space $W(\mathcal{F} L^1,L^\infty)$; in particular they could have stronger oscillations at infinity such as $\cos |ξ|^2$. Actually our main result deals with the more general case of possibly unbounded second derivatives. In that case we have boundedness on weighted modulation spaces with a sharp loss of derivatives.
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.