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  "body": "\n## Summary\n\nA band-limited function is a function on $\\R$ whose [[Fourier transform|DPVM34ADZV]] vanishes outside a bounded interval of frequencies $[-B, B]$.\nSuch functions are very rigid: by the [[Paley–Wiener theorem|THY12GPZ8C]] they extend to entire functions, so they are determined by their values on any interval.\nThey can be recovered from samples taken at a rate of $2B$ per unit time.\nThis is the content of the sampling theorem.\nSee the [[definition|B4TYSQQC38]].\n\n## Explanations\n\n### L1 human\n\nA band-limited signal contains no frequencies above some limit $B$.\nSuch a signal cannot change too fast, so it is pinned down by its values at evenly spaced times, $2B$ of them per second.\nThe function $\\sin(\\pi x)/(\\pi x)$ is the basic example: its spectrum is a box on $[-\\frac12, \\frac12]$.\nAudio recordings are filtered to be nearly band-limited before they are sampled.\n",
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