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  "body": "\n## Summary\n\nThe Fourier series of a periodic function writes it as a sum of sines and cosines, or of complex exponentials $e^{inx}$.\nIts coefficients are computed by integrals.\nIt turns problems about periodic functions into problems about sequences of coefficients.\nWhether and in which sense the series converges back to the function is a central question of harmonic analysis.\n[[Parseval's identity|047RFCNVSK]] and [[Fejér's theorem|TPJZCSXSVN]] give two answers.\n\n## Explanations\n\n### L1 human\n\nA musical sound can be split into a fundamental tone and its overtones.\nThe Fourier series does the same for any periodic function: it is a sum of pure waves whose frequencies are whole multiples of a base frequency.\nEach coefficient says how much of one wave is present.\nFor example, the sawtooth function equal to $x$ on $(-\\pi, \\pi)$ is $2(\\sin x - \\frac{1}{2}\\sin 2x + \\frac{1}{3} \\sin 3x - \\cdots)$ at every point strictly between $-\\pi$ and $\\pi$.\n",
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