Kolmogorov 1926: an $L^1$ function whose Fourier series diverges everywhere

There is $f \in L^{1}(\mathbb{T})$ whose Fourier series diverges at every point: for every $x \in \mathbb{T}$,

$\limsup_{N \to \infty} \big| S_N f(x) \big| = +\infty, \qquad S_N f(x) = \sum_{|n| \le N} \hat{f}(n) e^{inx}.$

Kolmogorov proved almost everywhere divergence in 1923 and everywhere divergence in 1926. The result is sharp in the scale of $L^p$ spaces: by Carleson's theorem (1966) and Hunt's extension (1968), the Fourier series of a function in $L^{p}(\mathbb{T})$ with $1 < p \le \infty$ converges almost everywhere, so $p = 1$ is exactly where everywhere divergence becomes possible.

Not proved in this library. It is recorded with citations and used in no proof here.

What would prove it. The construction is a lacunary sum of concentrated kernels, and it is genuinely hard: the difficulty is in arranging the partial sums to blow up at every point simultaneously, not merely on a large set. Beyond the construction, the statement itself cannot even be made here, since it quantifies over $L^{1}(\mathbb{T})$ and uses Fourier coefficients defined by a Lebesgue integral (Lebesgue measure and the Lebesgue integral ‡).

Which page it serves. The same future Fourier series page as du Bois-Reymond: a continuous function whose Fourier series diverges at a point ‡, and any future $L^p$ page. The two results are the boundary markers of the subject: continuity does not give pointwise convergence anywhere in particular, and $L^{1}$ membership does not give it anywhere at all, while $L^p$ for $p > 1$ gives it almost everywhere.

Attribution. The 1923 paper in Fundamenta Mathematicae gives divergence almost everywhere, and the 1926 note in the Comptes Rendus gives it everywhere. Both are Kolmogorov's, and the everywhere result is the one recorded above; only the 1923 paper has a freely readable scan, which is the link given.