We prove a sharp quantitative version of the Faber–Krahn inequality for the shorttime Fourier transform (STFT). To do so, we consider a deficit δ(f ;) which measures by how much the STFT of a function f ∈ L2(R) fails to be optimally concentrated on an arbitrary set ⊂ R2 of positive, finite measure. We then show that an optimal power of the deficit δ(f ;) controls both the L2-distance of f to an appropriate class of Gaussians and the distance of to a ball, through the Fraenkel asymmetry of . Our proof is completely quantitative and hence all constants are explicit. We also establish suitable generalizations of this result in the higher-dimensional context.

Stability of the Faber-Krahn inequality for the short-time Fourier transform / Gómez, Jaime; Guerra, André; Ramos, João P. G.; Tilli, Paolo. - In: INVENTIONES MATHEMATICAE. - ISSN 0020-9910. - STAMPA. - 236:2(2024), pp. 779-836. [10.1007/s00222-024-01248-2]

Stability of the Faber-Krahn inequality for the short-time Fourier transform

Tilli, Paolo
2024

Abstract

We prove a sharp quantitative version of the Faber–Krahn inequality for the shorttime Fourier transform (STFT). To do so, we consider a deficit δ(f ;) which measures by how much the STFT of a function f ∈ L2(R) fails to be optimally concentrated on an arbitrary set ⊂ R2 of positive, finite measure. We then show that an optimal power of the deficit δ(f ;) controls both the L2-distance of f to an appropriate class of Gaussians and the distance of to a ball, through the Fraenkel asymmetry of . Our proof is completely quantitative and hence all constants are explicit. We also establish suitable generalizations of this result in the higher-dimensional context.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11583/2987315