Let us consider function $f(x) \in C^{\infty}(-\infty, +\infty)$ that is $2\pi$-periodic on $\mathbb{R}$ (really $C^{\infty}(-\infty, +\infty)$ does not matter maybe just for simplicity). Consider its Fourier series $\displaystyle f(x) = \sum\limits_{k=-\infty}^{+\infty} a_k e^{ikx} = \sum\limits_{k=0}^{\infty} (A_k \sin{kx} + B_k \cos{kx})$ and denote $\displaystyle f_n(x) = \sum\limits_{k=-n}^{n} a_k e^{ikx} = \sum\limits_{k=0}^{n} (A_k \sin{kx} + B_k \cos{kx})$.
Do you know any examples of nontrivial functions $f(x)$ such that $f_n(x)$ are elementary functions?
In this case "nontrivial" means that it has an infinity Fourier series. Of course for example for $\sin{x}$ or $\cos{x}$ $f_n(x)$ are elementary, but it is a degenerate case.
My attempts were in the application of the Dirichlet kernel. So $\displaystyle f_n(x) = \frac{1}{2\pi}\int\limits_{-\pi}^{\pi} f(x+u)\frac{\sin((n + \frac{1}{2})u)}{2\sin{\frac{u}{2}}}du$. If we can find a function for which the integral can be computed in elementary functions then we solved the problem. But I got stuck in finding such a function.
Thanks for any help or ideas!