For a function $f$ from the Schwartz space $\mathcal{S}(\mathbb{R})$ do we have that  

$$\sum\limits\_{k\in\mathbb{Z}}\int\_{\mathbb{R}}|f(x+k)f'(x)|dx$$
converges?



Proof of $\sum\limits_{k\in\mathbb{Z}}\int_{\mathbb{R}}|f(x+k)f'(x)|dx<\infty$ for Schwartz function $f$
Trivially, you have the pointwise bound


$$\sum\_{k\in\mathbb{Z}} |f(x+k)f'(x)| \leq |f'(x)|\sup\_{y\in\mathbb{R}} \sum\_{k\in\mathbb{Z}} |f(y+k)|$$


By translation invariance,
$$\sup\_{y\in\mathbb{R}}\sum\_{k\in\mathbb{Z}} |f(y+k)| \leq \sup\_{y\in [0,1)} \sum\_{k\in\mathbb{Z}} |f(y+k)|,$$
and since $f$ is Schwartz by assumption,
$$\sup\_{y\in [0,1)} \sum\_{k\in\mathbb{Z}} |f(y+k)| \lesssim \sum\_{k\in\mathbb{Z}} (1+|k|)^{-2} <\infty.$$
Since $f$ is Schwartz, $f'\in L^1(\mathbb{R}))$. So by monotone convergence theorem, it follows that
$$\sum\_{k\in\mathbb{Z}} \int\_{\mathbb{R}} |f(x+k)f'(x)|dx <\infty.$$


