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

• 成果类型: Article
• 署名作者: Nicola, Fabio; Tilli, Paolo
• 署名单位: Polytechnic University of Turin
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-022-01119-8
• 发表日期: 2022
• 页码: 1-30
• 关键词: uncertainty principles

摘要

In this paper we solve an open problem concerning the characterization of those measurable sets Omega subset of 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 is an element of L-2(R-d) is defined in terms of its Short-time Fourier transform (STFT) nu f (x, omega), with Gaussian window. More precisely, given a measurable set Omega subset of R-2d having measure s > 0, we prove that the quantity Phi(Omega) = max { integral(Omega) vertical bar nu f(x,omega)vertical bar(2) dxd omega : f is an element of L-2(R-d), parallel to f parallel to L-2 = 1}, 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) <= 1 - e(-vertical bar Omega vertical bar), 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 is an element of [2, infinity), as well as to L-p-concentration estimates when p is an element of [1, infinity), thus proving a related conjecture. In all cases we identify the corresponding externals.