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?!

There is the following theorem of de la Vallée-Poussin, which I quote from page 880 of J. Marshall Ash's Uniqueness of Representation by Trigonometric Series:

If $S=\sum_n t_n e^{inx}$ converges to $f$ at each $x$,  and if $f$ is finite at each $x$ and if $\int_{\mathbb T} |f(x)| dx< \infty$, then $S$ is the Fourier series of $f$.

This theorem is to be found in Zygmund's Trigonometric Series, Vol I, page 326. The beginning of Ash's paper proves the special case for $f=0$, but it seems that the proof of the general case spans several pages, so I will leave it at that.
For your question: consider a continuous function $f$ constructed via the Uniform Boundedness Principle (see for example, the note of Paul Garrett linked in this question). Suppose that $S$ converged pointwise to $f$. By de la Vallée-Poussin's theorem, $S$ must be the Fourier series of $f$. But the Fourier series of $f$ does not converge at every point, which is a contradiction.
