Recent developments on Hardy's uncertainty principle
Seminari di Analisi Matematica dell'Università e del Politecnico di Torino
Date: Tuesday, 22 October 2024 14:30 - 15:30
Venue: Gianluca Giacchi - Aula C (Dipartimento di Matematica, Università di Torino)
Hardy's uncertainty principle is a foundational result in harmonic analysis, stating that a function in $L^2(\mathbb{R}^d)$ and its Fourier transform cannot both decay arbitrarily fast at infinity. Metaplectic operators, which generalize the Fourier transform, play a crucial role in various fields, including time-frequency analysis, quantum harmonic analysis, PDEs and signal analysis. These operators are strictly related to symplectic matrices, raising the question of whether their properties can be inferred directly from the structure of the associated symplectic matrices, or their "projections."
In this talk, we focus on the validity of Hardy's uncertainty principle for metaplectic operators. We demonstrate that every "non-trivial" metaplectic operator satisfies Hardy's uncertainty principle. Previously, this result was only established for metaplectic operators with "free" projections, which are straightforward extensions of the Fourier transform. However, we reveal that in general, Hardy's uncertainty principle exhibits a directionality that is not apparent in these simpler cases. This directionality is closely tied to the invertibility of a specific submatrix of the projection.
Speaker
Gianluca Giacchi
Affiliation
Università di Bologna
Room
Aula C
Proposed by
Elena Cordero
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