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  "body": "\n## Statement\n\nLet $g \\in L^2(\\R)$, and let $\\hat g(\\xi) = \\int_\\R g(t) e^{-2\\pi i t \\xi} \\, dt$ be its [[Fourier transform|DPVM34ADZV]].\nSuppose that the [[Gabor system|M9WHY8E2PM]] $\\left(e^{2\\pi i n t} g(t - k)\\right)_{k, n \\in \\Z}$ is a [[Riesz basis|ZYDYD7FB1Z]] of $L^2(\\R)$.\nThen\n$$\\left( \\int_\\R \\abs{t \\, g(t)}^2 \\, dt \\right) \\left( \\int_\\R \\abs{\\xi \\, \\hat g(\\xi)}^2 \\, d\\xi \\right) = \\infty.$$\n\n## Remarks\n\nSo a window of a Gabor Riesz basis at the critical density $ab = 1$ cannot be well localized both in time and in frequency.\nRoger Balian and, independently, Francis Low stated the result for orthonormal bases; later work completed their proofs and extended it to Riesz bases.\nA standard route uses the [[Zak transform|S026HQSGY9]]: the system is a Riesz basis exactly when $Zg$ is bounded above and below by positive constants almost everywhere.\nOn the other hand $Zg$ is quasi-periodic, and a continuous quasi-periodic function must vanish somewhere; when $g$ is well localized this contradicts the lower bound.\nThe hypothesis $ab = 1$ matters: for $ab \u003c 1$ the Gaussian $e^{-\\pi t^2}$, which is as well localized as possible, generates a Gabor frame.\n\nExample: $g = 1_{[0, 1)}$ generates an orthonormal basis.\nHere $\\int \\abs{t g(t)}^2 \\, dt = \\tfrac13$ is finite, but $\\abs{\\hat g(\\xi)} = \\abs{\\sin(\\pi \\xi)}/(\\pi \\abs{\\xi})$, so $\\abs{\\xi \\hat g(\\xi)}^2 = \\sin^2(\\pi \\xi)/\\pi^2$ is not integrable.\n\n## Explanations\n\n### L1 human\n\nA Gabor basis builds every signal from shifted and modulated copies of one window, placed on a grid as sparse as possible.\nThe theorem says that this window cannot be nice: it must be badly spread out either in time or in frequency.\nThe box function is sharp in time but its spectrum decays too slowly.\nIt is a strong form of the uncertainty principle for bases.\n\n### L2 human\n\nThe theorem explains why practical Gabor expansions are redundant frames with $ab \u003c 1$ rather than bases.\nIt is proved with the [[Zak transform|S026HQSGY9]], which turns the Gabor system at $a = b = 1$ into multiplication by a quasi-periodic function, and the question becomes one about zeros of that function.\nIt contrasts with wavelets: there exist orthonormal wavelet bases, such as the Meyer wavelets, whose generator is smooth and decays fast together with its Fourier transform.\nCompare with the [[uncertainty principle in Fourier analysis|S1FB2K14NV]], which bounds the product of the two integrals from below for one function.\n",
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