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 fin 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 ΦΩ=max{∫Ω|Vf(x,ω)|2dxdω:f∈L2(Rd),‖f‖L2=1},\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\Phi _\\Omega =\\max \\Big \\{\\int _\\Omega |{\\mathcal {V}}f(x,\\omega )|^2\\,dxd\\omega : f\\in L^2({\\mathbb {R}}^d),\\ \\Vert f\\Vert _{L^2}=1\\Big \\}, \\end{aligned}$$\\end{document}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 pin [2,infty ), as well as to L^p-concentration estimates when pin [1,infty ), thus proving a related conjecture. In all cases we identify the corresponding extremals.