Uniform convergence of generalised Fourier series

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.

• I'm not quite sure where the quantifiers go in your question; but, for example, if we fix some point $p \in S^1$, then the set of smooth functions vanishing at $p$ is dense in $L^2$, so we can find an orthonormal basis $u_n$ consisting entirely of such functions. If $u(p) \ne 0$ then the series will not even converge pointwise. May 16, 2019 at 3:40
• Thanks for your answer Nate. I tried to make a similar idea work, but ran into problems establishing the existence of a smooth orthonormal base of such functions (as opposed to merely $L^2$). Is this obvious? May 16, 2019 at 3:46
• Let $E$ be the set of such functions. Being a subset of the separable metric space $L^2$, $E$ is separable, so we can find a sequence $v_1, v_2, \dots$ which is dense in $E$ and therefore also dense in $L^2$. Now apply Gram-Schmidt to the sequence $v_n$. The resulting sequence $u_n$ will be orthonormal, and its linear span will be the same as that of the $v_n$, which is dense in $L^2$. May 16, 2019 at 3:48
• Ah of course, this was a silly question. Thank you. If you make that comment an answer I can accept it. May 16, 2019 at 3:52

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 $$u(x)$$.