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

Abstract

AbstractWe prove a sharp quantitative version of the Faber–Krahn inequality for the short-time Fourier transform (STFT). To do so, we consider a deficit $\delta (f;\Omega )$ δ ( f ; Ω ) which measures by how much the STFT of a function $f\in L^{2}(\mathbb{R})$ f ∈ L 2 ( R ) fails to be optimally concentrated on an arbitrary set $\Omega \subset \mathbb{R}^{2}$ Ω ⊂ R 2 of positive, finite measure. We then show that an optimal power of the deficit $\delta (f;\Omega )$ δ ( f ; Ω ) controls both the $L^{2}$ L 2 -distance of $f$ f to an appropriate class of Gaussians and the distance of $\Omega $ Ω to a ball, through the Fraenkel asymmetry of $\Omega $ 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.