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  "body": "\n## Statement\n\nLet $n \\ge 1$ and let $f \\in L^1(\\R^n)$.\nLet $\\hat f(\\xi) = \\int_{\\R^n} f(x) e^{-2\\pi i x \\cdot \\xi}\\, dx$ be its [[Fourier transform|DPVM34ADZV]].\nSuppose that $f$ vanishes almost everywhere outside a bounded set and that $\\hat f$ vanishes outside a bounded set.\nThen $f = 0$ almost everywhere.\n\n## Remarks\n\nIf $f$ vanishes outside a ball of radius $R$, then the integral $\\int f(x) e^{-2\\pi i x \\cdot z}\\, dx$ makes sense for every $z \\in \\C^n$.\nIt defines an [[entire function|0G4F0XB9X2]] of $z$ that extends $\\hat f$.\nAn entire function that vanishes on a nonempty open subset of $\\R^n$ vanishes identically, so $\\hat f = 0$, and then $f = 0$ by injectivity of the transform.\n\nThe statement is the simplest of a family of results that say a nonzero function and its Fourier transform cannot both be sharply concentrated.\nThe Heisenberg inequality bounds the product of the variances of $\\abs{f}^2$ and $\\abs{\\hat f}^2$ from below.\nHardy's uncertainty principle is about Gaussian decay, and the entropic uncertainty principle measures spread by entropy.\nBenedicks' theorem replaces bounded sets by sets of finite measure.\n\nExample: $\\operatorname{rect}$, the indicator of $[-\\frac12, \\frac12]$, has compact support, and its transform $\\sin(\\pi\\xi)/(\\pi\\xi)$ has zeros only at the nonzero integers, so it does not vanish outside any bounded set.\n\n## Explanations\n\n### L1 human\n\nA signal cannot be both short in time and limited to a finite band of frequencies.\nIf it lasts only a finite time, it must contain arbitrarily high frequencies, and if all its frequencies lie in a bounded band, it must go on forever.\n\n### L2 human\n\nThe proof uses the Paley–Wiener idea that compact support in $x$ makes $\\hat f$ analytic, indeed entire of exponential type.\nAnalyticity is very rigid, and this rigidity is the source of all qualitative uncertainty principles.\nIn signal processing the statement says that no nonzero band-limited signal has finite duration, so every real filter or sampling scheme is a compromise between the two.\n",
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