The space of periodizable tempered distribution

The periodization operator Per is defined for a Schwartz function φS(R) as
Per{φ}(x)=nZφ(xn),xR.
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
Per:OC(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

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