Coordinatable continuous functions in terms of orthogonal systems $\{e^{in\theta}\}$ Question. Let $f$ be an arbitrary  $2\pi$-periodic integrable function on reals. Does there exists any sequence $\{t_n\}$  with $f(\theta)=\sum t_ne^{in\theta}$? 
If NOT, what about if we replace integrable function with a continuous one?!

If a sequence $\{t_n|n\in\Bbb Z\}$ exists for which $f$ satisfies the identity$$f(\theta)=\sum_{n\in\Bbb Z}t_ne^{in\theta},$$applying $g\mapsto\int_0^{2\pi}g(\theta)e^{-ik\theta}d\theta$ with $k\in\Bbb Z$ gives $\int_0^{2\pi}f(\theta)e^{-ik\theta}d\theta=2\pi t_k$, since in terms of the Kronecker delta $\int_0^{2\pi}e^{i(n-k)\theta}d\theta=2\pi\delta_{nk}$ for integers $n,\,k$. In other words, if a solution exists it is$$f(\theta)=\sum_{n\in\Bbb Z}\left(\frac{1}{2\pi}\int_0^{2\pi}f(\theta^\prime)e^{-in\theta^\prime}d\theta^\prime\right)e^{in\theta}.$$If such an $f$ is modified at countably many values of $\theta$, the integrals in the second display-line equation are unchanged, so the function obtained from such a sum is the "original" function. Then the display-line equations are each true of the "new" $f$, except at countably many points.
