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  "body": "\n## Summary\n\nHarmonic analysis on groups studies functions on a locally compact group by writing them as superpositions of basic oscillations attached to the group.\nFor abelian groups the oscillations are the characters and the main tool is the Fourier transform on the dual group.\nFor non-abelian groups the characters are replaced by irreducible unitary representations.\nThe subject contains Fourier series, the Fourier transform on $\\R^n$, and the representation theory of compact and Lie groups.\n\n## Explanations\n\n### L1 human\n\nA musical sound can be split into pure tones; harmonic analysis does the same for functions.\nWhich \"pure tones\" are available depends on the symmetry of the space: on a circle they are the waves $e^{in\\theta}$, on the line they are $e^{i\\xi x}$.\nOn a sphere, where rotations act, the pure tones are spherical harmonics, and they come from representations of the rotation group.\nHarmonic analysis on groups is the general study of this splitting for any group with a reasonable topology.\n",
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