Stability of the Faber-Krahn inequality for the Short-time Fourier Transform
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2023-07-18
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Working Paper
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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 $δ(f;Ω)$ which measures by how much the STFT of a function $f\in L^2(\mathbb R)$ fails to be optimally concentrated on an arbitrary set $Ω\subset \mathbb R^2$ of positive, finite measure. We then show that an optimal power of the deficit $δ(f;Ω)$ controls both the $L^2$-distance of $f$ to an appropriate class of Gaussians and the distance of $Ω$ to a ball, through the Fraenkel asymmetry of $Ω$. 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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published
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Pages / Article No.
2307.09304
Publisher
Cornell University
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Edition / version
v1
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Subject
Classical Analysis and ODEs (math.CA); Analysis of PDEs (math.AP); Functional Analysis (math.FA); FOS: Mathematics
Organisational unit
09565 - Figalli, Alessio / Figalli, Alessio
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Funding
721675 - Regularity and Stability in Partial Differential Equations (EC)
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