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  "body": "\n## Summary\n\nFourier analysis is the study of how functions are represented by sums or integrals of sines, cosines and complex exponentials.\nIts main tools are the [[Fourier series|G12KXDHPPS]] of periodic functions, the [[Fourier transform|DPVM34ADZV]] on $\\R^n$ and the [[discrete Fourier transform|TST0GH66A5]] on $\\Z / N\\Z$.\nIt is the classical core of [[harmonic analysis|7QNRC9SJVV]].\nIt turns differentiation into multiplication and [[convolution|2Z29JN3VES]] into pointwise product, which makes it central for differential equations and signal processing.\n\n## Explanations\n\n### L1 human\n\nFourier analysis splits a signal into its frequencies, much as a prism splits white light into colours.\nA periodic signal is a sum of waves whose frequencies are whole multiples of one base frequency.\nA non-periodic signal needs all frequencies, and the sum becomes an integral.\nFor example, the function $\\cos^2 x$ equals $\\frac{1}{2} + \\frac{1}{2} \\cos 2x$, so it contains only the frequencies $0$ and $2$.\nJoseph Fourier used such decompositions to solve the [[heat equation|VP2HN86VAG]].\n",
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