Fourier transform of $2\pi$-periodic function I want to describe the image of $2\pi$-periodic tempered distributions under the Fourier transform
$$F: \mathcal S'(\mathbb R) \to \mathcal S'(\mathbb R), f \mapsto \left(\varphi \mapsto \langle F[f](\varphi), \varphi \rangle := \langle f, F[\varphi]\rangle\right),$$
where $F[\varphi](\xi) = \frac {1} {\sqrt{2 \pi}}\displaystyle\int_{-\infty}^\infty e^{-i x \xi} \varphi(x) dx$ for $\varphi \in \mathcal S(\mathbb R)$, i.e. I'm trying to describe explicitly the set $F[\mathcal T] \subset \mathcal S'(\mathbb R)$, where $\mathcal T := \{f \in \mathcal S'(\mathbb R): f(x + 2\pi) = f(x)\}$.
I managed to show that if $f \in \mathcal T$, then $g = F[f] \in F[\mathcal T]$ must satisfy $(1 - e^{-2 \pi i \xi}) g(\xi) = 0$, from which I deduced that $\mathrm{supp}(g) \subset \mathbb Z$, and moreover
$$g(\xi) = \sum_{k \in \mathbb Z} c_k \delta(\xi - k).$$
Now I cannot manage what are the restrictions on coefficients $c_k$. I have a hypothesis that
$$F[\mathcal T] = \{\sum_{k \in \mathbb Z} c_k \delta(\xi - k) : |c_k| \le C \cdot (1 + |x|^m) \text{ for some } C, m\},$$
but I'm stuck trying to prove it. Any help would be very appreciated.
 A: Claim. $\mathcal F[\mathcal T] = \{\sum_{k \in \mathbb Z} c_k \delta(\xi - k): |c_k| \le C (1 + |k|^m) \text{ for some } C > 0, m \in \mathbb Z_+\}$.
$\fbox{$\subseteq$}$
For $g \in \mathcal D'(\mathbb R^n)$ the following are equivalent:

*

*$g \in \mathcal S'(\mathbb R^n)$;

*there exist $A, B > 0, \alpha \in \mathbb Z_+^n$ such that $|\langle g, \varphi \rangle| \le A \sup_{\xi \in \mathbb R^n} |(1 + |\xi|^B) D^\alpha \varphi(\xi)|$ for all $\varphi \in \mathcal D(\mathbb R^n)$.

Since $g = \mathcal F[f] \in \mathcal S'$, such $A, B > 0$ and $\alpha \in \mathbb Z_+$ do actually exist.
Then for each $k\in\mathbb Z$ there is a test function $\varphi_k \in \mathcal D(\mathbb R)$ with $\mathrm{supp}(\varphi_k) \subset B_1(k), \varphi_k(k) = 1$ and $|\varphi_k^{(\alpha)}(\xi)| \le \kappa$ for $\xi \in \mathbb R$ where $\kappa = \kappa(\alpha) > 0$ is some constant.
Consider some $g(\xi) = \sum_{k \in \mathbb Z} c_k \delta(\xi - k) \in \mathcal F[\mathcal T]$. If $g \notin \mathrm{rhs}$, then there is some $k \in \mathbb Z_+$, such that $|c_k| > A \kappa (\lceil B \rceil + 1)2^{\lceil B \rceil}(1 + |k|^{\lceil B \rceil}) \ge A \cdot \kappa (1 + (|k| + 1)^B)$.
Then the following inequalities couldn't be satisfied at the same time:

*

*$|\langle g, \varphi_k \rangle| = |c_k \varphi_k(k)| > A \cdot \kappa (1 + (|k| + 1)^B)$;

*$|\langle g, \varphi_k \rangle| \le A \sup_{B_1(k)} |(1 + |\xi|^B) \varphi_k^{(\alpha)}(\xi)| \le A \cdot \kappa (1 + (|k| + 1)^B)$.

$\fbox{$\supseteq$}$
It is easy to see that $g(x) = \sum_{k \in \mathbb Z} c_k \delta(x - k) \in \mathcal S'(\mathbb R)$ if $|c_k| \le C(1 + |k|^m)$ for some $C > 0$ and $m \in \mathbb Z_+$.
Note that $\mathcal F^{-1}[\delta(x - k)] = \frac 1{\sqrt{2\pi}} \exp(ik\xi)$ is $2\pi$-periodic function. Hence, $f(\xi) := \mathcal F^{-1}[g] \in \mathcal T$ and the inclusion follows.
