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  "body": "\n## Statement\n\nLet $\\sigma \u003e 0$, and let $F$ be an [[entire function|0419794R0T]].\nThe following two conditions are equivalent.\n\n1. $F$ is of [[exponential type|9P8D8YAHQS]] at most $\\sigma$, and its restriction to the real line is square-integrable: $\\int_{-\\infty}^{\\infty} \\abs{F(x)}^2 \\, dx \u003c \\infty$.\n2. There is a square-integrable function $\\varphi$ on the interval $[-\\sigma, \\sigma]$ such that\n$$F(z) = \\int_{-\\sigma}^{\\sigma} \\varphi(t) \\, e^{i z t} \\, dt \\quad \\text{for all } z \\in \\C.$$\n\n## Remarks\n\nThe implication from 2 to 1 is direct: $\\abs{e^{izt}} = e^{-t \\operatorname{Im} z} \\le e^{\\sigma \\abs{z}}$ for $\\abs{t} \\le \\sigma$, and the Plancherel theorem gives the square-integrability on $\\R$.\nFor the converse, $\\varphi$ is the Fourier transform of $F$ on $\\R$, up to a constant.\nIt vanishes outside $[-\\sigma, \\sigma]$; this follows by shifting the line of integration into the upper or lower half-plane.\nThe growth there is controlled by the [[Phragmén–Lindelöf principle|FYR6V2NH6C]].\nFor example, with $\\varphi = 1/2$ on $[-\\sigma, \\sigma]$ one gets $F(z) = \\sin(\\sigma z)/z$, which has exponential type $\\sigma$ and is square-integrable on $\\R$.\nThe square-integrability is needed: $\\sin(\\sigma z)$ has exponential type $\\sigma$ but is not of the form 2, since it is not square-integrable on $\\R$.\n\n## Explanations\n\n### L1 human\n\nA signal whose frequencies all lie in a bounded band $[-\\sigma, \\sigma]$ extends from the real line to an entire function of the complex variable.\nThat entire function grows at most like $e^{\\sigma \\abs{z}}$.\nThe theorem says that the converse holds too: such growth, together with finite energy on the real line, forces the frequencies into the band.\nSo the smallest band $[-\\sigma, \\sigma]$ that contains the frequencies is given by the exponential type.\n\n### L2 human\n\nThe space of functions in the theorem is the Paley–Wiener space, a Hilbert space of entire functions that is the basic example of a [[de Branges space|NYHNTP9AND]].\nIt is the setting of the sampling theorem of signal processing: such a function is determined by its values at the points $\\pi n/\\sigma$, $n \\in \\Z$.\nThere are related Paley–Wiener theorems for smooth functions and distributions with compact support, where the growth condition also involves $\\abs{\\operatorname{Im} z}$.\n",
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