Elenco seminari del ciclo di seminari
“SOME LIKE IT FOUIRER”

It is well-known that the Fourier transform of the characteristic function of a set is able to detect the "singularities" of the set, or more in general its geometric properties. In this talk I would like to review some results at the interface between geometric measure theory and harmonic analysis. In particular I would like to propose a notion of s-mass for surfaces which has a spectral nature, and recovers the classical mass when s tends to 1 from below.

The high-frequency behaviour of the Fourier transform of a set encodes the geometry of its boundary. For instance, the spherical averages of |ℱ(1_Ω)|² decay like R⁻ᵈ⁻¹ for convex domains and for domains with C^{1,1/2} boundary (Brandolini–Hofmann–Iosevich). With L. Gennaioli, we proved that for bounded BV functions the averaged high-frequency energy converges exactly to the L²-energy of the jumps. For indicator functions of sets, the same limit yields the perimeter. Luc Tartar asked whether the Besov space B^{1/2}_{2,∞}, defined by a similar condition on high-frequencies, could replace BV for hyperbolic equations with discontinuities. That space, however, is too large for jumps to behave well. I will discuss the problem of finding an intermediate space, ideally described on the frequency side, where the jump identity and the nice geometric features of BV survive.

The aim of the talk is to introduce some notions of the theory of involutive structures. This allows us to study systems of complex vector fields translating the solvability problems into the language of differential complexes and their cohomology, making it easier to apply tools from algebraic topology and differential geometry. It includes many classical examples, such as the de Rham complex, the Dolbeault complex, and the tangential Cauchy-Riemann complex. When the underlying manifold is a Lie group, the analytical problem of solvability of systems of vector fields is strongly connected to the algebraic properties of the Lie algebra and sometimes it can be determined by them.

Yes. In this talk I will present an existence theorem for sets of given area that maximize the sum of the first eigenvalues of localization operators. In particular, I will present the results in the "concrete" setting of the Fock space (where optimal sets are known to be balls), and later discuss the extension to localization operators associated with an abstract wavelet transform satisfying some mild assumptions. This is based on joint work with Fabio Nicola and Paolo Tilli.

We establish anisotropic uncertainty principles (UPs) for general metaplectic operators acting on L²(ℝᵈ), 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 rank(B). First, we prove sharp Heisenberg-Pauli-Weyl type inequalities involving only the directions corresponding to ker(B)⊥, 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-Hörmander 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.

Imposing too strong simultaneous decay or localization constraints on a function and its Fourier transform causes the function to vanish identically (Uncertainty Principle). The same applies to a solution to some dispersive evolutionary PDE if similar constraints are imposed on its profile at two different time points. In the talk, I will first show a manifestation of the Uncertainty Principle, the Amrein-Berthier-Benedicks Theorem. The L^2 norm on R^n of a function is controlled by the sum of the L^2 norms of the function and its Fourier transform observed outside sets of finite Lebesgue measure (Amrein-Berthier Theorem). In particular, no non-zero function supported on a set of finite Lebesgue measure can have a Fourier transform with the same property (Benedicks Theorem). Then, I will show how the above statements translate into qualitative and even quantitative unique continuation results when the Fourier transform is replaced by the propagator, evaluated at a non-zero time T, associated to some dispersive PDE, in particular considering linear and non-linear instances of the Schrödinger equation.

Inspired by edge detection based on the decay behavior of wavelet coefficients, we introduce a linear-time algorithm for detecting the local regularity in non-uniformly sampled multivariate signals. Our approach quantifies regularity within the framework of microlocal spaces introduced by Jaffard. The central tool in our analysis is the fast samplet transform, a distributional wavelet transform tailored to scattered data. We establish a connection between the decay of samplet coefficients and the pointwise regularity of multivariate signals. While traditional wavelets are effective for regularity detection in low-dimensional structured data, they are not suited to process irregularly sampled or scattered data. On the contrary, samplets demonstrate robust performance even for higher dimensional and scattered data.

Graph signal processing provides a mathematical framework for the analysis of data defined on networks and other irregular domains, and the graph Fourier transform, defined through the eigenvectors of the graph Laplacian, extends classical Fourier analysis to signals defined on graphs [2]. More generally, graph signals may be observed at several time instants without assuming a specific evolution law. This has motivated the development of joint time-vertex signal processing methods, which combine temporal analysis with the spectral structure of the graph. In particular, Loukas and Foucard [3] introduced a joint time-vertex Fourier transform, while Grassi et al. [1] developed a framework for joint filtering and signal representation. In this work, we study a path-extension model for time-varying graph signals, obtained by attaching a temporal path to each vertex of a weighted undirected graph, with the spatial connections retained only at the initial layer. The Laplacian of the enlarged graph admits a Kronecker representation that reduces its spectral analysis to a family of smaller transition matrices. We characterize their eigenvalues and eigenvectors through trigonometric and hyperbolic equations and establish a connection with Chebyshev polynomials of the third kind. This spectral decomposition allows us to define a graph Fourier transform in which the temporal basis depends on the spatial frequencies, unlike in the standard Cartesian product construction. We also develop an accelerated numerical procedure for computing the eigenvalues and representing the eigenvectors without constructing the full Laplacian matrix. References [1] F. Grassi, A. Loukas, N. Perraudin, B. Ricaud, A time vertex signal processing framework: Scalable processing and meaningful representations for time-series on graphs, IEEE Transactions on Signal Processing, vol. 66, no. 3, pp. 817–829, 2018. [2] D. K. Hammond, P. Vandergheynst, R. Gribonval, Wavelets on graphs via spectral graph theory, Applied and Computational Harmonic Analysis, vol. 30, no. 2, pp. 129–150, 2011. [3] A. Loukas, D. Foucard, Frequency analysis of time-varying graph signals, Proceedings of the 2016 IEEE Global Conference on Signal and Information Processing (GlobalSIP), pp. 346–350, 2016.

TBA