Given a function $f(z)$, $z=x+iy, x,y\in \mathbb R$, which belongs to $\mathbb H^{2}(\mathbb C^{+})$, where $\mathbb C^{+}$ is the upper half plane Im$(z)>0$ and $f(a_{n})=0$, for all $n\in \mathbb Z$, where $a_{n}'s$ are all real numbers. What can we say about this function $f$? should it be the zero function!? or there is something else that we can say about?
Edit: $f$ is continuous on the real line, and has singularities in the lower half plane, and the sequence $\{a_{n}\}$ has no accumulation point.