Let $I \subset \mathbb R$ be a compact interval, $\mu$ is a complex regular Borel measure on $I$ (e.g. $\mu = g\,dt$ for some $g \in L^1(I)$, $dt$ the Lebesgue measure) and $g : \mathbb C \to \mathbb C$ an entire function. Define an entire function $F : \mathbb C \to \mathbb C$ via $$ F(z) = \int_I g(t-z) \, d\mu(t). $$ I'm interested in the zero set of such functions $F$, in particular in the question if one can make certain assumptions on $g$ so that a discrete set of the form $\mathbb Z$ (or similar sets) constitute a uniqueness set for $F$, i.e. $F(\mathbb Z) = 0 \implies F=0 $.

For instance, one could assume that $g$ is a function of exponential type with certain integrability assumptions and then use Shannon's sampling theorem. I'm searching for more general assumptions.

I was wondering if someone of you came across such problems or knows papers/studies about it. Thanks in advance!