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

Question: Can we define a proper subspace of $\mathcal{S}(\mathbb{R})$ 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"?

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  • $\begingroup$ Would this just be the set of functions $\{f: \forall x\in\Bbb{R}, \sum\limits_{n\in\Bbb{Z}} f(x-n)<\infty\}$? $\endgroup$
    – Ryan Hendricks
    Commented Aug 16, 2020 at 19:35
  • $\begingroup$ @fierydemon First I also want to include generalized functions with no pointwise definition (e.g., the Dirac comb $\sum_{n\in\mathbb{Z}}\delta(\cdot - n)$. But even with well-defined pointwise values, it is not obvious to me what is a good function space to deal with. In your suggestion, what is the meaning of the infinite sum? Pointwise convergence for each $x$? Where is the function $f$? These are typically the questions I am facing. $\endgroup$
    – Goulifet
    Commented Aug 16, 2020 at 21:22

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