This is an problem from Folland's Real Analysis
Let $E_k(x)=e^{2\pi k \cdot x}$. If $g:\mathbb Z^n\to \mathbb C $ satisfies $|g(k)|\leq C(1+|k|)^N$ for some $C, N>0$, then the series $\sum_{k\in\mathbb Z^n}g(k)E_k$ converges in $\mathcal D'(\mathbb T^n)$ to a distribution $F$ that satisfies $\hat F=g$. It also converges in $\mathcal S'(\mathbb R^n)$ to a tempered distribution $G=P'F$ such that $\tau_kG=G$ for all $k$.
Here the periodization map $P$ is defined as $Pf=\sum_{k\in\mathbb Z^n}\tau_k\phi$. I managed to show that $P$ is continuous from $\mathcal S(\mathbb R^n)$ to $C^\infty (\mathbb T^n)$. Therefore $P'$ is also continuous.
But I am having trouble to show the convergence of the series $\sum_{k\in\mathbb Z^n}g(k)E_k$ in $\mathcal D'(\mathbb T^n)$ .
Any help will be appreciated.