Let $f \in L^2([0,1])$ . Then Carleson's Theorem states that

$$\lim_{N\to \infty} \sum_{|n|<N} \langle f,e_n\rangle e_n(x)=f(x),\quad\text{a.e. } x\in[0,1],$$ where $\{e_n\}$ is the Orthonormal basis of $L^2([0,1])$ defined by $e_n(x)=e^{2\pi in x}$ and $\langle\,\cdot\,,\,\cdot\,\rangle$ is the usual inner product of $L^2[0,1]$ defined as $$\langle f,g\rangle:=\int_0^1 f(x)\overline {g(x)} dx.$$

Now my question is: what makes the particular orthonormal base $\{e^{2\pi i nx}\}$ so special?

For which Orthonormal basis $\{w_n\}$ of $L^2[0,1]$, can we say that for every $f \in L^2[0,1]$, that $$\lim_{N\to \infty} \sum_{|n|<N} \langle f,w_n\rangle w_n(x)=f(x),\quad\text{a.e. } x\in[0,1]?$$

And what if we ask the same question with $L^2$ replaced everywhere by $C[0,1]$ ?