 Du Bois-Reymond theorem

The Du Bois-Reymond theorem says that continuity of a periodic function does not guarantee convergence of its ordinary Fourier series at every point. At any prescribed circle point $x_0$, there is even a real continuous $f$ with unbounded Fourier partial sums $S_{N}f(x_0)$.

For $x_0=0$, view $T_N:f\mapsto S_{N}f(0)$ as bounded linear functionals on the real Banach space $C(\mathbb{T})$ with its supremum norm. The Dirichlet kernel harmonic lower bound givesIf every continuous input had bounded values under all $T_N$, the Uniform boundedness principle would bound these operator norms, a contradiction. Translation moves the exceptional point to any specified $x_0$. The generic unbounded Fourier sums at a fixed point strengthens this existence proof to a category statement.^[ref]

This does not contradict mean-square convergence: a continuous function is square-integrable, and its series still converges in $L^2$. Nor does it obstruct Fejér's theorem, which averages partial sums and gives uniform convergence for every continuous periodic input. The Fejér kernel is positive and its sums are a uniform-norm contraction, but the sums themselves need not be nonnegative: for $f=-1$, every Fejér sum equals $-1$.