@article{StabilitySTFT,
	abstract = {We prove a sharp quantitative version of the Faber--Krahn inequality for the short-time Fourier transform (STFT). To do so, we consider a deficit {\$}{$\backslash$}delta (f;{$\backslash$}Omega ){\$}which measures by how much the STFT of a function {\$}f{$\backslash$}in L\^{}{\{}2{\}}({$\backslash$}mathbb{\{}R{\}}){\$}fails to be optimally concentrated on an arbitrary set {\$}{$\backslash$}Omega {$\backslash$}subset {$\backslash$}mathbb{\{}R{\}}\^{}{\{}2{\}}{\$}of positive, finite measure. We then show that an optimal power of the deficit {\$}{$\backslash$}delta (f;{$\backslash$}Omega ){\$}controls both the {\$}L\^{}{\{}2{\}}{\$}-distance of {\$}f{\$}to an appropriate class of Gaussians and the distance of {\$}{$\backslash$}Omega {\$}to a ball, through the Fraenkel asymmetry of {\$}{$\backslash$}Omega {\$}. Our proof is completely quantitative and hence all constants are explicit. We also establish suitable generalizations of this result in the higher-dimensional context.},
	author = {G{\'o}mez, Jaime and Guerra, Andr{\'e} and Ramos, Jo{\~a}o P. G. and Tilli, Paolo},
	date = {2024/05/01},
	doi = {10.1007/s00222-024-01248-2},
	isbn = {1432-1297},
	journal = {Inventiones mathematicae},
	number = {2},
	pages = {779--836},
	title = {Stability of the Faber-Krahn inequality for the short-time Fourier transform},
	url = {https://doi.org/10.1007/s00222-024-01248-2},
	volume = {236},
	year = {2024},
}