Pointwise a.e. Convergence of Fourier Series in $L^p$ Space?

Question

Does the Fourier Series of functions in ${L^p}$(T) converge pointwise a.e. to the functions themselves (T here is [-$\pi, \pi$] with 2$\pi$ periodicity)? I know in $L^2$(T) the functions $e^{inx}$ for integer n forms an orthonormal basis of $L^2$(T), so every $L^2$ function has a convergent Fourier Series equal to itself, but not sure about the general $L^p$ case.

Answer 1

The fact that the exponentials are an orthonormal basis for $L^2$ does not at all directly imply anything about pointwise convergence, only $L^2$ convergence. But, as @uniquesolution notes, L. Carleson proved that in fact the Fourier series of $L^2$ functions do converge pointwise almost everywhere. Then R. Hunt proved (just a few years later) the corresponding pointwise a.e. result for $L^p$ functions.

Answer 2

Yes, it does. It is a very deep theorem due to Carleson. It was probably the most difficult open question about Fourier Series, until Carleson settled it in around $1966$, if I am not mistaken.

https://en.wikipedia.org/wiki/Carleson's_theorem