Ciro Santilli @cirosantilli 34

The Fourier series of an $L^2$ 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 $L^2$, 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.

Riesz-Fischer theorem is a norm version of it, and Carleson's theorem is stronger pointwise almost everywhere version.

Note that the Riesz-Fischer theorem is weaker because the pointwise limit could not exist just according to it: $L^p$ norm sequence convergence does not imply pointwise convergence.

$L^p$ is:

And then this is why quantum mechanics basically lives in $L^2$: not being complete makes no sense physically, it would mean that you can get closer and closer to states that don't exist!

TODO intuition