Fourier series
Convergence

A proof that a Fourier series is a valid representation of any periodic function (that satisfies the Dirichlet conditions) is overviewed in § Fourier theorem proving convergence of Fourier series.

In engineering applications, the Fourier series is generally assumed to converge except at jump discontinuities since the functions encountered in engineering are better-behaved than functions encountered in other disciplines. In particular, if s {\displaystyle s} is continuous and the derivative of s (x) {\displaystyle s(x)} (which may not exist everywhere) is square integrable, then the Fourier series of s {\displaystyle s} converges absolutely and uniformly to s (x) {\displaystyle s(x)}. If a function is square-integrable on the interval {\displaystyle }, then the Fourier series converges to the function at almost everywhere. It is possible to define Fourier coefficients for more general functions or distributions, in which case point wise convergence often fails, and convergence in norm or weak convergence is usually studied.
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