Convergence of Fourier series 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?
 A: *

*Convergence in $L^p$, $p>1$.
True, by M. Riesz's Theorem. This is a standard topic in every harmonic analysis course, with several readable proofs.


*Convergence pointwise almost everywhere, $p>1$.
True, by the Carleson-Hunt Theorem. 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.


*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.


*Convergence pointwise almost everywhere, $p=1$.
False, due to Kolmogorov 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.
