Say $f \in L^p[a,b]$, with $p \in \mathbb{N}, p > 1 $. Does its Fourier Series converge in the metric space $L^p[a,b]$? Does the series converge pointwise? And at which conditions?
Say now $p = 1$, Does its Fourier Series converge in the metric space $L^1[a,b]$? Does the series converge pointwise? And under which conditions?



Convergence of Fourier series
1. **Convergence in $L^p$, $p>1$.**


True, by [M. Riesz's Theorem](https://www.jstor.org/stable/1993749). This is a standard topic in every harmonic analysis course, with several readable proofs.


2. **Convergence pointwise almost everywhere, $p>1$.**


True, by the [Carleson-Hunt Theorem](https://terrytao.wordpress.com/2020/05/14/247b-notes-4-almost-everywhere-convergence-of-fourier-series/#more-11764). This result is over 50 years old, but famously difficult. Though the techniques used are now standard, there aren't really any easy proofs, even for continuous functions.


3. **Convergence in $L^1$.**


False, and not very difficult to prove. Suppose that Fourier series converged in $L^1$. Let $S\_N$ be the operator that takes a (Schwartz, say) function and returns the $N$th partial Fourier sum. I claim that these operators are not bounded uniformly in the $L^1$ norm, which is enough to give a contradiction by the uniform boundedness principle. For indeed applying it to the Fejer kernel $K\_M$ gives $$\|S\_NK\_M\|\_{L^1}\to\|D\_N\|\_{L^1}$$ as $M\to\infty$. Where $D\_N$ is the Dirichlet kernel. But $\|D\_N\|\_{L^1}$ is unbounded, so $S\_N$ cannot be uniformly bounded.


4. **Convergence pointwise almost everywhere, $p=1$.**


False, due to [Kolmogorov](https://iopscience.iop.org/article/10.1070/RM1983v038n04ABEH004205) almost 100 years ago. In fact, it is possible to find an $L^1$ function whose Fourier series diverges *everywhere*. This is not an easy counterexample, but it is presented in some courses in harmonic analysis.


