TITLE: When I can safely assume that a function is a Laplace transform of other function?
QUESTION [18 upvotes]: If I have a function and I want to represent it as being the Laplace transform of another, that is, I want to be sure that there is $\hat{f}(s)$ such that my function $f(x)$ can be written as:
$$f(x) = \int ds \hat{f}(s) \exp(-sx)$$
what conditions should I impose over $f(x)$?
In other words, what are the conditions for the Fourier–Mellin–Bromwich integral
$$\hat{f}(s) = \frac{1}{2\pi i} \int_{\gamma - i\infty}^{\gamma + i\infty} f(x) \exp(sx) dx$$
to exist?

REPLY [8 votes]: This kind of question is very interesting, and I too would like to know answers.
Sorry to self-publicise; I hope it's not regarded as impolite, but since I have also considered this exact kind of question, it's quickest just to refer to my own paper (and the references I give in there): Laplace Transform Representations and Paley–Wiener Theorems for Functions on Vertical Strips by Zen Harper, Documenta Math. 15 (2010) 235-254.
Given an analytic function on a vertical strip, I try to find conditions which guarantee it can be represented by a bilateral Laplace transform (in various senses). I definitely don't claim to have any kind of complete answer, but it's the best I know of (by definition! If I knew any better answers, I would have written them in my paper!).
It seems like there are still many open questions about this.