Carleson's theorem
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The Fourier series of an function (i.e. the function generated from the infinite sum of weighted sines) converges to the function pointwise almost everywhere.
The theorem also seems to hold (maybe trivially given the transform result) for the Fourier series (TODO if trivially, why trivially).
Only proved in 1966, and known to be a hard result without any known simple proof.
This theorem of course implies that Fourier basis is complete for , as it explicitly constructs a decomposition into the Fourier basis for every single function.
TODO vs Riesz-Fischer theorem. Is this just a stronger pointwise result, while Riesz-Fischer is about norms only?
One of the many fourier inversion theorems.