Let $f : \mathbb R_+ \to \mathbb C$ be a measurable and integrable function where $\mathbb R_+ = [0,\infty)$. The Laplace transform of $f$ is given by $$ Lf(s) = \int_0^\infty f(x)e^{-sx} \, dx. $$ A classical Theorem due to Lerch [M. Lerch, Sur un point de la théorie des fonctions génératrices d’Abel, Acta Math. 27, 339 -351 (1903)] states that if $$ Lf(\delta n) = 0, \quad n \in \mathbb N $$ for some $\delta>0$ then $f=0$ almost everywhere. Suppose now that we replace the Laplace transform by the two-sided Laplace transform, $$ Tf(s) = \int_{\mathbb R} f(x)e^{-sx} \, dx. \quad (*) $$ Suppose that $Tf$ defines an entire function (clearly, we need a decay assumption on the negative axis so that $Tf$ is entire).

**Question:** Are there Lerch-type results for the two-sided Laplace transform? That is, suppose that $f$ belongs to a certain function class $C \subset L^1(\mathbb R)$. Can we find discrete sets $A \subset \mathbb R$, so that $Tf(a)$ for all $a \in A$ implies $f=0$ almost everywhere? In particular, I'm interested in function classes $C$ so that $A$ can be chosen to be uniformly discrete, $\inf_{a \neq b, a,b \in A} > 0$.