Schwartz functions after periodization has period 1

Question

We define periodization of a function $f$ as $$ P_1f(x) = \sum_{n\in N} f(x+n) = \lim_{N\rightarrow \infty} \sum_{|n|\leq N} f(x+n). $$ To show that $P_1f$ has period 1, my idea was to do a change of variable so that $m=n+1$. However, this does not use any property of the Schwartz class and it seems applicable to any integer form, which looks wrong to me. Any help?

Answer 1

The only property of a Schwartz function used here is that $$\tag{*}\left(1+(x+n)^2\right)\lvert f(x+n)\lvert\leqslant C,$$ which guarantees that $P_1f(x)$ is defined for every $x$ and makes all the computations with series valid.

Inequality $(*)$ comes from the fact that for each $k\geqslant 0$, there exists a constant $C_k$ such that $\sup_{t\in\mathbb R}\lvert t\rvert^k\lvert f(t)\rvert\leqslant C_k$; here we take $C:=C_0+C_2$.