Remark

There exist functions $f\in L^1(\mathbf{T})$ whose Fourier series diverges at every point $x\in \mathbf{T}$ (Kolmogorov's theorem, cf. exercise 51 of II, p. 289).

A theorem of Carleson[^1] shows that the symmetric partial sums of the Fourier series of $f$ converge to $f(x)$ for almost all $x\in \mathbf{T}$ if $f\in \mathscr{L}^2(\mathbf{T})$.