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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.
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The theorem also seems to hold (maybe trivially given the transform result) for the Fourier series (TODO if trivially, why trivially).
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Only proved in 1966, and known to be a hard result without any known simple proof.
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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.
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TODO vs Riesz-Fischer theorem. Is this just a stronger pointwise result, while Riesz-Fischer is about norms only?
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Ancestors

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