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.



Uniform convergence of generalised Fourier series
The series need not converge uniformly, nor even pointwise everywhere.


For instance, fix $x \in S^1$ and consider the space $C^\infty\_x$ of smooth functions that vanish at $x$. This is a dense subspace of $L^2(S^1)$, so by choosing a countable dense subset of $C^\infty\_x$ and applying Gram-Schmidt, we can find a sequence $u\_1, u\_2, \dots \in C^\infty\_x$ which is an orthonormal basis for $L^2(S^1)$. Then if $u(x) \ne 0$, the series evaluated at $x$ converges to 0, not to $u(x)$.


