du Bois-Reymond: a continuous function whose Fourier series diverges at a point

There is a continuous $2\pi$-periodic function $f : \mathbb{R} \to \mathbb{R}$ whose Fourier series diverges at a point: writing

$S_N f(x) = \sum_{|n| \le N} \hat{f}(n) e^{inx}, \qquad \hat{f}(n) = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(t) e^{-int} \, dt,$

there is $x_0$ with $\limsup_{N} |S_N f(x_0)| = +\infty$. du Bois-Reymond gave the first such example in 1873. The set of points of divergence can be taken to be any prescribed set of measure zero, and by Carleson's theorem it can be no larger than that: the Fourier series of a continuous, indeed of any $L^2$, function converges almost everywhere.

Not proved in this library, for now. Unlike most of this page, this result is probably reachable here, and it is recorded rather than proved only because the page that would carry it has not been written.

What would prove it. Two routes. The explicit one is du Bois-Reymond's lacunary construction, a sum of blocks of conjugate Dirichlet kernels with rapidly increasing frequencies, which is elementary but intricate. The soft one is the uniform boundedness principle applied to the functionals $f \mapsto S_N f(0)$ on $C(\mathbb{T})$, whose norms are the Lebesgue constants $\|D_N\|_{L^1} \sim (4/\pi^2)\log N \to \infty$; the Baire category theorem then gives a comeagre set of continuous functions whose Fourier series diverge at $0$. The Baire category theorem for complete metric spaces is in scope in this library, and uniform boundedness is flagged in the deferral list as borderline, since its proof is Baire plus linearity. So the soft route becomes available as soon as normed spaces and the Lebesgue constants exist.

Which page it serves. A Fourier series page in the analysis track, which is not yet planned, and the approximation and compactness page, where the positive results live: Fejer's theorem gives uniform convergence of the Cesaro means for every continuous function, and Weierstrass approximation follows. The correct reading of the pair is that summability, not convergence, is the right notion for continuous functions.

Attribution. The first published proof is du Bois-Reymond, Ueber die Fourierschen Reihen, Nachr. Kon. Ges. Wiss. Gottingen 21 (1873), 571-582, which is the paper the cited Wikipedia article names as the first proof that the Fourier series of a continuous function can diverge. The extended treatment is his 1876 Munich memoir, and 1876 is quoted for the result by part of the literature; the earlier date is used here because it is the date of the first publication.

Contrast. This is the continuous, pointwise, Baire-reachable failure. The $L^1$ failure recorded in Kolmogorov 1926: an $L^1$ function whose Fourier series diverges everywhere ‡ is of a different order: it is everywhere and it needs the Lebesgue theory even to state.