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  "body": "\n## Definition\n\nLet $f \\in L^2(\\R)$, and let $\\hat f \\in L^2(\\R)$ be its [[Fourier transform|DPVM34ADZV]], normalized so that $\\hat f(\\xi) = \\int_\\R f(x) e^{-2\\pi i x \\xi}\\, dx$ for integrable $f$.\nLet $B \u003e 0$.\nThe function $f$ is *band-limited* with bandwidth $B$ if $\\hat f(\\xi) = 0$ for almost every $\\xi$ with $\\abs{\\xi} \u003e B$.\nIt is *band-limited* if it is band-limited with some bandwidth $B \u003e 0$.\n\n## Remarks\n\nIf $f$ is band-limited with bandwidth $B$, then $\\hat f$ lies in $L^2[-B, B] \\subset L^1[-B, B]$, and $f$ agrees almost everywhere with the continuous function $x \\mapsto \\int_{-B}^{B} \\hat f(\\xi) e^{2\\pi i x \\xi}\\, d\\xi$; one always uses this continuous representative.\n\nExample: $f(x) = \\sin(\\pi x)/(\\pi x)$, with $f(0) = 1$, lies in [[$L^2(\\R)$|SASP14MA1Z]], and $\\int_{-1/2}^{1/2} e^{2\\pi i x \\xi}\\, d\\xi = f(x)$.\nSo $\\hat f$ is the indicator of $[-\\frac12, \\frac12]$, and $f$ is band-limited with bandwidth $\\frac12$.\n\nNon-example: $f(x) = e^{-\\pi x^2}$ has $\\hat f(\\xi) = e^{-\\pi \\xi^2}$, which is positive everywhere, so $f$ is not band-limited.\n\nBy the [[Paley–Wiener theorem|THY12GPZ8C]], the band-limited functions with bandwidth $B$ are exactly the restrictions to $\\R$ of certain entire functions.\nThese are the entire functions of exponential type at most $2\\pi B$ that are square-integrable on $\\R$.\nIn particular a nonzero band-limited function cannot vanish on an interval.\nThe same definition is used for tempered distributions, with the condition that the support of the distribution $\\hat f$ lies in $[-B, B]$.\nIn this wider sense $\\sin(2\\pi B x)$ is band-limited, though not in $L^2$.\n",
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    "name": "Band-limited function",
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