The space of periodizable tempered distribution

The periodization operator Per is defined for a Schwartz function φS(R) as
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 OC(R) (see for instance this paper for details). We then have
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”?


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