We know by the Hausdorff-Bernstein-Widder theorem that any completely monotonic function on the positive half line $[0, \infty)$ is given by the Laplace transform of a positive Borel measure on $[0, \infty)$, and vice versa. A natural question is how can we characterize the space of functions which are the difference $f_1 - f_2$ of two completely monotonic functions $f_1, f_2$. By the same theorem, we know that these functions are equivalent to space of functions which are Laplace transforms of all signed Borel measures. I cannot find any results studying this space and whether, for example, this space of functions can approximate all decreasing functions, all bounded functions, all functions of bounded variation, all functions of a given exponential order, etc.
I'd be really interested to understand how expressive this space is and whether there are any known approximation results of what functions can be approximated by differences of completely monotonic functions.
