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  "body": "\n## Summary\n\nThe inverse Fourier transform takes a function of frequency and builds a function of position by superposing the waves $e^{2\\pi i x \\cdot \\xi}$ with these amplitudes.\nIt has the same form as the [[Fourier transform|DPVM34ADZV]], with the opposite sign in the exponent.\nThe [[Fourier inversion theorem|T4Z8YQZKZV]] says that it undoes the Fourier transform, so a function can be recovered from its frequency content.\n\n## Explanations\n\n### L1 human\n\nThe Fourier transform measures how much of each frequency a signal contains.\nThe inverse Fourier transform goes back: given the amount of each frequency, it adds up all the corresponding waves to rebuild the signal.\nThe two operations look almost the same, with only a sign changed in the exponent.\nFor example, the bell curve $e^{-\\pi \\abs{\\xi}^2}$ is sent to the same bell curve $e^{-\\pi \\abs{x}^2}$.\n",
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