Stability of the Faber-Krahn inequality for the Short-time Fourier Transform

TL;DR

This paper establishes a precise quantitative relationship between the concentration deficit of the short-time Fourier transform of a function and its proximity to Gaussian functions, as well as the shape of the concentration set, with explicit constants.

Contribution

It provides the first sharp quantitative stability result for the Faber-Krahn inequality applied to the STFT, including explicit constants and higher-dimensional generalizations.

Findings

The deficit controls the $L^2$-distance to Gaussian functions.

The deficit bounds the asymmetry of the concentration set.

Results are quantitative with explicit constants.

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 $\delta(f;\Omega)$ 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 $\Omega\subset \mathbb R^2$ of positive, finite measure. We then show that an optimal power of the deficit $\delta(f;\Omega)$ controls both the $L^2$-distance of $f$ to an appropriate class of Gaussians and the distance of $\Omega$ to a ball, through the Fraenkel asymmetry of $\Omega$. Our proof is completely quantitative and hence all constants are explicit. We also establish suitable generalizations of this result in the higher-dimensional context.