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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.
The Fourier series behaves really nicely in , where it always exists and converges pointwise to the function: Carleson's theorem.
Video 2. But what is a Fourier series? From heat flow to circle drawings | DE4 by 3Blue1Brown (2019) Source. Amazing 2D visualization of the decomposition of complex functions.