The periodization operator Per is defined for a Schwartz function φ∈S(R) as

Per{φ}(x)=∑n∈Zφ(x−n),∀x∈R.

The sum in (1) is of course well-defined pointwise due to the rapid decay of φ and we then have that Per{φ} is an infinitely smooth 1-periodic function. More generally, it is possible to define the periodization operator Per over rapidly decaying distributions O′C(R) (see for instance this paper for details). We then have

Per:O′C(R)→S′(T)

continuously, the latter space being the space of 1-periodic distributions.

Question:Can we define a proper subspace of S(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”?

**Answer**

**Attribution***Source : Link , Question Author : Goulifet , Answer Author : Community*