Kolmogorov's everywhere divergent Fourier series

Remarks

Kolmogorov first gave, in 1923, an integrable function whose Fourier series diverges almost everywhere, and in 1926 one whose Fourier series diverges everywhere. The construction adds up a series of carefully chosen trigonometric polynomials. Each has small $L^1$ norm, but some of its partial sums are very large on most of the circle. The coefficients $\hat f(k)$ still tend to $0$, by the Riemann–Lebesgue lemma, so the divergence is not caused by large terms. The example shows that Carleson's theorem and Hunt's extension, which give convergence almost everywhere for $f \in L^p(\mathbb{T})$ with $1 < p < \infty$, have no analogue for $p = 1$.