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  "body": "\n## Statement\n\nLet $f \\colon \\R \\to \\C$ be measurable, and let $a, b \u003e 0$ and $C \\ge 0$ be constants.\nSuppose that $\\abs{f(x)} \\le C e^{-\\pi a x^2}$ for all $x \\in \\R$, so that $f$ is integrable.\nLet $\\hat f(\\xi) = \\int_\\R f(x) e^{-2\\pi i x \\xi}\\, dx$ be the [[Fourier transform|DPVM34ADZV]] of $f$.\nSuppose that $\\abs{\\hat f(\\xi)} \\le C e^{-\\pi b \\xi^2}$ for all $\\xi \\in \\R$.\n\n1. If $ab \u003e 1$, then $f = 0$.\n2. If $ab = 1$, then $f(x) = c\\, e^{-\\pi a x^2}$ for some constant $c \\in \\C$.\n\n## Remarks\n\nBy the [[Fourier transform of the Gaussian|ZNEHMR2CCP]], $e^{-\\pi a x^2}$ has transform $a^{-1/2} e^{-\\pi \\xi^2 / a}$, so case 2 cannot be improved.\nIf $ab \u003c 1$, then $e^{-\\pi c x^2}$ satisfies both bounds, with a suitable constant $C$, for every $c$ with $a \\le c \\le 1/b$, so there are nonzero functions of this kind.\n\nThe proof uses complex analysis.\nThe decay of $f$ makes $\\hat f$ extend to an entire function with $\\abs{\\hat f(\\xi + i\\eta)} \\le C' e^{\\pi \\eta^2 / a}$.\nIn case 2, the entire function $g(z) = e^{\\pi z^2 / a} \\hat f(z)$ is bounded on the real and on the imaginary axis.\nThe [[Phragmén–Lindelöf principle|FYR6V2NH6C]], applied in each quadrant, shows that $g$ is bounded, hence constant by Liouville's theorem, and Fourier inversion gives $f$.\nCase 1 reduces to case 2 with $b$ replaced by $1/a$, and then the bound $\\abs{\\hat f(\\xi)} \\le C e^{-\\pi b \\xi^2}$ with $b \u003e 1/a$ forces the constant to be $0$.\n\nThe result is due to G. H. Hardy.\nThere are versions on $\\R^n$, and versions with $L^p$ bounds in place of pointwise bounds.\n\n## Explanations\n\n### L1 human\n\nThe bell curve $e^{-\\pi x^2}$ and its transform decay at the same Gaussian rate.\nHardy's theorem says that no nonzero function can do better on both sides at once.\nIf both $f$ and $\\hat f$ decay faster than this matched pair of Gaussians, the function must be zero.\n\n### L2 human\n\nUnlike the Heisenberg inequality, which measures spread with variances, Hardy's theorem measures concentration through pointwise decay.\nIt is a sharp qualitative statement, and the borderline case singles out the Gaussians, just as the equality case of Heisenberg's inequality does.\nIts proof is a model for the use of Phragmén–Lindelöf arguments in Fourier analysis.\nIt has been extended to Lie groups and to the Schrödinger equation, where it gives unique continuation results.\n",
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