Consider a function $f(t)$ that is only defined via its Laplace transform $F(s)$
$f(t) := \int\limits_{c - i\infty}^{c + i \infty} d s \; e^{st} F(s)$,
where $c$ is chosen appropriately (i.e. to the right-hand side of all of $F(s)$'s poles, yaddayadda).
Given $F(s)$, I want to know the positions $t_0$ of $f(t)$'s extremal points, that is the position of the maxima and minima. I want to find $t_0$ such that $f(t_0) = 0$.
Of course, I know about the differentiation rule $f'(t) \mapsto s F(s) - f(0)$, but that helps me only so far.
Are there any theorems on stuff like that. Is there hope...?
Thanks in advance.
PS: I actually don't care so much about the Laplace-transform. It could also be some general integral transform of a measure $dF(s)$.
EDIT 1: After putting some more thought into the question, I realized that it is of course sufficient to ask for the zeros of a function defined by its Laplace transform. Furthermore, when parametrizing the integration contour with $s = c + i k$ , $k \in [-\infty,\infty]$, and redefining $\tilde{F}(k) := e^{ct} F(c + i k)$, one can map the question to "How to obtain the zeros of a function defined by its Fourier transform?" I.e. how to find $t_0$ such that:
$0 = f(t_0) = \int\limits_{-\infty}^{\infty} dk e^{ikt_0}\tilde{F}(k)$
Looking at the absolute value of $f(t)$, one can get rid of the complex exponential. However, this leads to the condition that $\tilde{F}(k) = 0$ almost surely, which implies that $f(t)$ is identically zero. This does not help. Writing down the real and imaginary parts of the integral separately and setting them to zero, gives an equivalent condition on the cosine and sine transforms of the real and imaginary parts of $\tilde{F}(k)$. But after that I do not know how to contiinue...
I guess there are at least results on the absense of zeros. Where can I read about them. (E.g. In probability theory, one needs to ensure from the characteristic function, that the original pdf is non-negative...)