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  "body": "\n## Summary\n\nA modulation space $M^{p,q}(\\R^d)$ is a space of tempered distributions defined by the size of their [[short-time Fourier transform|3DHNP5NEWH]].\nThe exponent $p$ measures integrability in the time variable and $q$ in the frequency variable.\nModulation spaces are the natural function spaces of [[time–frequency analysis|GQFPVXEWK1]].\nThey are used to study pseudo-differential operators, Gabor expansions and dispersive equations.\n\n## Explanations\n\n### L1 human\n\nA piece of music is described by which notes sound at which times.\nThe short-time Fourier transform of a signal gives such a description: a number for each time $x$ and each frequency $\\omega$.\nA modulation space collects the signals whose time–frequency picture has finite size in a mixed norm.\nFor $p = q = 2$ the size is the usual energy, and the space is $L^2(\\R^d)$.\nA Gaussian pulse lies in every modulation space, since its time–frequency picture is again a Gaussian bump.\n",
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