Suppose $u_n$ is an orthonormal basis of smooth functions on $S^1$.

Does there exist a smooth function $u$ such that the generalised Fourier series $$u=\sum_{n\in\mathbb{N}} \langle u,u_n\rangle u_n $$ does not converge uniformly?

We of course have uniform convergence for the standard Fourier basis by integration by parts, but I cannot find a counterexample (or a proof of uniform convergence) for the general case.