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Let $I \subset \mathbb R$ be a compact interval, $f \in L^1(I)$ and $g : \mathbb C \to \mathbb C$ an entire function. Define an entire function $F : \mathbb C \to \mathbb C$ via $$ F(z) = \int_I g(t-z) f(t) \, dt. $$ 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!

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    $\begingroup$ If $\mu=g\,dt$ and $ F(z) = \int_I g(t-z) \, d\mu(t),$ then why not write $ F(z) = \int_I g(t-z) g(t)\,dt$ and dispense with the notation involving $\mu \text{ ?} \qquad$ $\endgroup$
    – Michael Hardy
    Commented Jan 8, 2022 at 17:10
  • $\begingroup$ you're right, I changed it, thanks! $\endgroup$
    – Muzi
    Commented Jan 8, 2022 at 17:50
  • $\begingroup$ In your special example, I do not think it is important that $f$ is compactly supported since you are just using the fact that $g$ is band limited. If $F$ decays quickly enough you may still apply the Poisson summation formula for cases other than your special case. But without the band limited assumption for $F$ you are not recovering $\widehat{F}$ from the sample of $F$ on $\mathbb{Z}$. $\endgroup$
    – Shannon Starr
    Commented Jan 9, 2022 at 20:52

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