Let $f(x)=\sum_{n=-\infty}^{\infty} c_ne^{inx}$ in $\mathcal{S}'(\mathbb{R})$. To show: $|c_n|\le M|n|^m$ for $|n|\ge 1$, some $M>0$ and $m\in \mathbb{N}$.
We have $\forall \phi\in \mathcal{S}(\mathbb{R}):\langle f(x)-\sum_{n=-N}^N c_ne^{inx},\phi(x)\rangle\to 0, $ whenever $N\to \infty$. So $\forall \phi\in\mathcal{S}(\mathbb{R})$: $$\forall \varepsilon>0: \exists K\in\mathbb{N}: \forall N\ge K: \left|\langle \sum_{n=-N}^N c_ne^{inx},\phi(x)\rangle-\langle f(x),\phi(x)\rangle\right|<\varepsilon.$$
Fix some $\phi$, then we may assume that the support of $\phi$ is contained in $[-A,A]$. Then $$ \langle \sum_{n=-N}^N c_ne^{inx},\phi(x)\rangle = \sum_{n=-N}^N c_n\int_{-A}^A e^{inx}\phi(x)dx.$$
In particular $\langle f,\phi\rangle$ is finite. How do I get closed to the required upper bounds?
Thanks.
Take $\psi \in C^\infty_c, \int \psi=1$, let $\phi = \psi \ast 1_{x\in [0,2\pi]}$.
If $c_n$ isn't bounded by any polynomial then there is $a_n$ rapidly decreasing such that $a_n c_n$ isn't bounded.
Let $$\varphi(x) = \phi(x)\sum_n a_n e^{-inx} $$ $\varphi$ is $C^\infty_c$ and $$\langle \sum_{n=-N}^N c_n e^{inx},\varphi \rangle = 2\pi\sum_{n=-N}^N c_n a_n$$ which doesn't converge as $N\to \infty$, contradicting that $\lim_{N\to \infty} \sum_{n=-N}^N c_n e^{inx}$ converges in $S'(\Bbb{R})$.