# The space of periodizable tempered distribution

The periodization operator $$Per\mathrm{Per}$$ is defined for a Schwartz function $$φ∈S(R)\varphi \in \mathcal{S}(\mathbb{R})$$ as
$$Per{φ}(x)=∑n∈Zφ(x−n),∀x∈R.$$\mathrm{Per} \{ \varphi \} (x) = \sum_{n \in \mathbb{Z}} \varphi( x - n ), \quad \forall x \in \mathbb{R}. \tag{1}$$$$
The sum in (1) is of course well-defined pointwise due to the rapid decay of $$φ\varphi$$ and we then have that $$Per{φ}\mathrm{Per}\{\varphi\}$$ is an infinitely smooth $$11$$-periodic function. More generally, it is possible to define the periodization operator $$Per\mathrm{Per}$$ over rapidly decaying distributions $$O′C(R)\mathcal{O}_{C}'(\mathbb{R})$$ (see for instance this paper for details). We then have
$$Per:O′C(R)→S′(T)$$\mathrm{Per} : \mathcal{O}_{C}'(\mathbb{R}) \rightarrow \mathcal{S}'(\mathbb{T}) \tag{2}$$$$
continuously, the latter space being the space of $$11$$-periodic distributions.

Question: Can we define a proper subspace of $$S(R)\mathcal{S}(\mathbb{R})$$ that maximally extends the periodization in a precise sense? That is, a space on which the periodization is well-defined, with a natural topology that makes the periodization continuous, with good reasons for its “maximality”?