The Faber-Krahn inequality for the Short-time Fourier transform

TL;DR

This paper characterizes the sets in time-frequency space that maximize energy concentration of the Short-time Fourier transform, proving a Faber-Krahn type inequality and deriving sharp uncertainty principles and extremal functions.

Contribution

It solves an open problem by identifying extremal sets for energy concentration in the STFT and establishes a sharp uncertainty principle with explicit extremals.

Findings

Maximal energy concentration occurs on sets equivalent to a ball.

Derived a sharp bound for the STFT's essential support in one dimension.

Identified extremal functions achieving equality in the bounds.

Abstract

In this paper we solve an open problem concerning the characterization of those measurable sets $\Omega\subset \mathbb{R}^{2d}$ that, among all sets having a prescribed Lebesgue measure, can trap the largest possible energy fraction in time-frequency space, where the energy density of a generic function $f\in L^2(\mathbb{R}^d)$ is defined in terms of its Short-time Fourier transform (STFT) $\mathcal{V} f(x,\omega)$, with Gaussian window. More precisely, given a measurable set $\Omega\subset\mathbb{R}^{2d}$ having measure $s> 0$, we prove that the quantity \[ \Phi_\Omega=\max\Big\{\int_\Omega|\mathcal{V} f(x,\omega)|^2\,dxd\omega: f\in L^2(\mathbb{R}^d),\ \|f\|_{L^2}=1\Big\}, \] is largest possible if and only if $\Omega$ is equivalent, up to a negligible set, to a ball of measure $s$, and in this case we characterize all functions $f$ that achieve equality. This result leads to a sharp uncertainty principle for the "essential support" of the STFT (when $d=1$, this can be summarized by the optimal bound $\Phi_\Omega\leq 1-e^{-|\Omega|}$, with equality if and only if $\Omega$ is a ball). Our approach, using techniques from measure theory after suitably rephrasing the problem in the Fock space, also leads to a local version of Lieb's uncertainty inequality for the STFT in $L^p$ when $p\in [2,\infty)$, as well as to $L^p$-concentration estimates when $p\in [1,\infty)$, thus proving a related conjecture. In all cases we identify the corresponding extremals.