= Fourier series
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Approximates an original function by sines. If the function is "well behaved enough", the approximation is to arbitrary precision.
<Fourier>'s original motivation, and a key application, is <solving partial differential equations with the Fourier series>.
Can only be used to approximate for periodic functions (obviously from its definition!). The <Fourier transform> however overcomes that restriction:
* https://math.stackexchange.com/questions/1115240/can-a-non-periodic-function-have-a-fourier-series
* https://math.stackexchange.com/questions/1378633/every-function-can-be-represented-as-a-fourier-series
The Fourier series behaves really nicely in <l2>, where it always exists and converges pointwise to the function: <Carleson's theorem>.
\Video[https://www.youtube.com/watch?v=r6sGWTCMz2k]
{title=But what is a <Fourier series>? by <3Blue1Brown> (2019)}
{description=Amazing 2D visualization of the decomposition of complex functions.}