Let $A=\{ z \in \mathbb C : |\mathrm{Im}(z)| < a \}$ for some $a >0$. Further let $F:A \to \mathbb C$ be a holomorphic function on $A$ with the following properties:

- $F$ is bounded on $A$
- For every $y \in (-a,a)$ the map $x \mapsto |F(x + iy)|$ has a certain decay to zero as $|x| \to \infty$, for instance an exponential decay.

I'm wondering if for such functions certain uniqueness results holds true, in the sense that if $F$ vanishes on a certain discrete set on the real axis (say the integers $\mathbb Z$) then $F$ vanishes identically. Is anyone of you familiar with results in this direction?