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  "body": "\n## Summary\n\nHarmonic analysis studies functions by writing them as superpositions of basic waves, and studies operators through their action on these waves.\nIts classical core is the theory of [[Fourier series|G12KXDHPPS]] on the circle and of the [[Fourier transform|DPVM34ADZV]] on $\\R^n$.\nOn a general locally compact group the waves are replaced by characters or irreducible representations, which gives [[harmonic analysis on groups|5E6KB3V8R0]].\nThe subject supplies tools for partial differential equations, number theory, probability and signal processing.\n\n## Explanations\n\n### L1 human\n\nA chord played on a piano is a sum of single notes, and a trained ear can hear which notes are present.\nHarmonic analysis does the same for functions: it splits a function into pure oscillations and records how much of each frequency it contains.\nMany questions become easier on the side of the frequencies, because differentiation and translation act very simply on a pure wave.\nFor example, the derivative of $e^{inx}$ is $in \\, e^{inx}$, so differentiation becomes multiplication by $in$.\nThe hard part is to put the pieces back together and control in which sense the sum of the waves gives back the function.\n",
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