Let $w$ be a rapidly decaying function on $\mathbb{R}$ such that
$$ \sum\_{n \in \mathbb{Z}} w(x+n) = 0$$
for all $x \in \mathbb{R}$. Does that imply that $w$ is identically zero? What if we assume that $w$ is continuous?



Constant "periodization" of a function
No, the **Haar wavelet** is a counter example (i.e. $w(x) = \chi\_{[0,1[}(x) - \chi\_{[1,2[}(x)$). Decay is arbitrarily fast and mollifying by convolution gives a smooth counterexample.


