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  "body": "\n## Summary\n\nA Fourier multiplier is an operator that acts on a function by multiplying its [[Fourier transform|DPVM34ADZV]] by a fixed function $m$, called the symbol.\nDifferential operators with constant coefficients, the [[Hilbert transform|NSCEBXM7ZM]], the heat semigroup and frequency filters are all of this form.\nEvery symbol in $L^\\infty$ gives a bounded operator on $L^2$, and the bounded operators on $L^2(\\R^n)$ that commute with translations are exactly these.\nWhich symbols give bounded operators on $L^p$ for $p \\ne 2$ is a central and hard question of harmonic analysis, studied through [[$L^p$ multipliers|RAGR38Q40B]].\nSee the [[definition|SD89TZ1PHR]].\n\n## Explanations\n\n### L1 human\n\nAn equalizer on a music player raises or lowers each frequency by its own factor.\nA Fourier multiplier does exactly that to a function.\nIt splits the function into frequencies, multiplies the amount of frequency $\\xi$ by a number $m(\\xi)$, and puts the pieces back together.\nFor example, $m(\\xi) = 1$ for $\\abs{\\xi} \\le 1$ and $0$ otherwise is an ideal low-pass filter that keeps only the low frequencies.\n",
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