Let us denote $T$ by the unit circle. Let $\{e_n\}$ be an orthonormal basis for $L^2(T)$, with respect to Lebesgue measure.

We say $\{e_n\}$ is ** smooth** if it satisfies the following property:

$$f(t){=}\lim_{N\to \infty}\sum_{-N}^{N}\langle f,e_n\rangle e_n(t)~~~~~(\forall f\in C(T))$$

Q. Does there exists any smooth orthonormal basis for $L^2(T)$?