arXiv subjects

Edoardo Pucci

Publications and source records attributed to Edoardo Pucci.

5 recordsLinked to original sources

arXiv2026

Beyond Single-Window Graph Fourier Analysis

We introduce a multi-windowed graph Fourier transform (MWGFT) for the joint vertex-frequency analysis of signals defined on graphs. Building on generalized translation and modulation induced by the graph Laplacian, the proposed framework extends the windowed graph Fourier transform by allowing multiple analysis and synthesis windows. Exact reconstruction formulas are derived for complex-valued graph signals, together with sufficient and computable conditions guaranteeing stable invertibility. The associated families of windowed graph Fourier atoms are shown to form frames for the space of graph signals. Numerical experiments on synthetic and real world graphs confirm exact reconstruction up to machine precision and demonstrate improved stability and vertex-frequency localization compared to single-window constructions, particularly on irregular graph topologies.

math.CA↗

arXiv2026

Anisotropic uncertainty principles for metaplectic operators

We establish anisotropic uncertainty principles (UPs) for general metaplectic operators acting on $L^2(\mathbb{R}^d)$, including degenerate cases associated with symplectic matrices whose $B$-block has nontrivial kernel. In this setting, uncertainty phenomena are shown to be intrinsically directional and confined to an effective phase-space dimension given by $\mathrm{rank}(B)$. First, we prove sharp Heisenberg-Pauli-Weyl type inequalities involving only the directions corresponding to $\ker(B)^\perp$, with explicit lower bounds expressed in terms of geometric quantities associated with the underlying symplectic transformation. We also provide a complete characterization of all extremizers, which turn out to be partially Gaussian functions with free behavior along the null directions of $B$. Building on this framework, we extend the Beurling-H\"ormander theorem to the metaplectic setting, obtaining a precise polynomial-Gaussian structure for functions satisfying suitable exponential integrability conditions involving both $f$ and its metaplectic transform. Finally, we prove a Morgan-type (or Gel'fand--Shilov type) uncertainty principle for metaplectic operators, identifying a sharp threshold separating triviality from density of admissible functions and showing that this threshold is invariant under metaplectic transformations. Our results recover the classical Fourier case and free metaplectic transformations as special instances, and reveal the geometric and anisotropic nature of uncertainty principles in the presence of symplectic degeneracies.

math.AP↗

arXiv2025

$\mathcal{A}$-Localization Operators

Time-frequency localization operators, originally introduced by Daubechies (1988), provide a framework for localizing signals in the phase space and have become a central tool in time-frequency analysis. In this paper we introduce and study a broad generalization of these operators, called $\mathcal{A}$-localization operators, associated with a metaplectic Wigner distribution $W_\mathcal{A}$ and the corresponding $\mathcal{A}$-pseudodifferential calculus. We first show that the classical relation between localization operators and Weyl quantization extends to any \emph{covariant metaplectic Wigner distribution}. Specifically, if $W_\mathcal{A}$ satisfies the covariance property \[ W_\mathcal{A}(π(z)f,π(z)g)=T_zW_\mathcal{A}(f,g), \qquad z\in\mathbb{R}^{2d}, \] then \[ A_{a}^{φ_1,φ_2} = \operatorname{Op}_\mathcal{A}\big(a * W_\mathcal{A}(φ_2,φ_1)\big), \] and conversely, this identity characterizes covariance. This result extends the recent representation formula of Bastianoni and Teofanov for $τ$-operators to the full metaplectic framework. We then define the $\mathcal{A}$-localization operator $A_{a,\mathcal{A}}^{φ_1,φ_2}$ and investigate its analytical properties. We establish boundedness results on modulation spaces and provide sufficient conditions for Schatten-von Neumann class membership. These findings connect the structure of metaplectic representations with time-frequency localization theory, offering a unified approach to quantization and signal analysis.

math.FA↗

arXiv2025

Sparse Gabor representations of metaplectic operators: controlled exponential decay and Schrödinger confinement

Motivated by the phase space analysis of Schrödinger evolution operators, in this paper we investigate how metaplectic operators are approximately diagonalized along the corresponding symplectic flows by exponentially localized Gabor wave packets. Quantitative bounds for the matrix coefficients arising in the Gabor wave packet decomposition of such operators are established, revealing precise exponential decay rates together with subtler dispersive and spreading phenomena. To this aim, we present several novel results concerning the time-frequency analysis of functions with controlled Gelfand-Shilov regularity, which are of independent interest. As a byproduct, we generalize Vemuri's Gaussian confinement results for the solutions of the quantum harmonic oscillator in two respects, namely by encompassing general exponential decay rates as well as arbitrary quadratic Schrödinger propagators. In particular, we extensively discuss some prominent models such as the harmonic oscillator, the free particle in a constant magnetic field and fractional Fourier transforms.

math.AP↗

arXiv2024

Understanding of linear operators through Wigner analysis

In this work, we extend Wigner's original framework to analyze linear operators by examining the relationship between their Wigner and Schwartz kernels. Our approach includes the introduction of (quasi-)algebras of Fourier integral operators (FIOs), which encompass FIOs of type I and II. The symbols of these operators reside in (weighted) modulation spaces, particularly in Sjöstrand's class, known for its favorable properties in time-frequency analysis. One of the significant results of our study is demonstrating the inverse-closedness of these symbol classes. Our analysis includes fundamental examples such as pseudodifferential operators and Fourier integral operators related to Schr{ö}dinger-type equations. These examples typically feature classical Hamiltonian flows governed by linear symplectic transformations $S \in Sp(d, \mathbb{R})$. The core idea of our approach is to utilize the Wigner kernel to transform a Fourier integral operator $ T $ on $ \mathbb{R}^d $ into a pseudodifferential operator $ K$ on $ \mathbb{R}^{2d}$. This transformation involves a symbol $σ$ well-localized around the manifold defined by $ z = S w $.

math.AP↗