# analytic continuation of a Laplace transform from a countably infinite set of points?

Let $f(\lambda)=\int_0^\infty e^{-\lambda s} F(ds)$, where $F$ is the distribution of a positive random variable. Suppose I know the value of $f(n)$ for $n=0,1,2,\cdots$. Is this enough to uniquely determine $f(\lambda)$ in its domain of convergence (DOC)?

Since the Laplace transform is analytic in its DOC, knowledge of $f(\lambda$) on a line segment uniquely determines $f(\lambda)$ on its domain. But, knowledge of the value of an analytic function at a countably infinite number of points does not, in general, determine the function throughout its domain. I was hoping that, analyticity + the fact that $f(\lambda)$ is a Laplace transform would be enough to analytically continue $f(\lambda)$ from a countably infinite set of points.

It would seem like the fact that $f(\lambda)$ is positive, decreasing and convex might be useful.

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A uniqueness theorem you formulated is proved here: https://www.math.lsu.edu/~neubrand/ital.pdf for the case where $F$ satisfies the Lipschitz condition.