I think I read somewhere that if a Fourier series converges pointwise to some $f$ and if the Fourier coefficients are absolutely summable, then the Fourier series converges uniformly.
But now I am not sure that this is actually true.
Is this true?
What I do know is that if the limit function $f$ is continuous and the coefficients are absolutely summable then the convergence is uniform. But does this work with continuity replaced with pointwise convergence? My intuition tells me that this feels wrong.
Suppose $f$ is 1-periodic and $(c_n)$ is the sequence of its Fourier coefficients. Let $S_nf = \sum_{|k|\leq n} c_ke^{2\pi ik\cdot}$ be the $n$th partial sum. Then, $$\sup_x |S_nf(x)-S_mf(x)| \leq \sum_{n<|k|\leq m}|c_n|$$ for $n<m$. Thus, if the Fourier series of $f$ is absolutely summable, then $(S_nf)$ is a Cauchy sequence in $C(\mathbb{R}/\mathbb{Z},\|\cdot\|_\infty)$, so uniformly convergent.