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

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 \[ Φ_Ω=\max\Big\{\int_Ω|\mathcal{V} f(x,ω)|^2\,dxdω: f\in L^2(\mathbb{R}^d),\ \|f\|_{L^2}=1\Big\}, \] 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 ${\Phi }_{\Omega }\le 1-e^{-|\Omega |}$, 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.

23 pages

References in corpus (1)

• Spectral deviation of concentration operators for the short-time Fourier transform

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