I am interesting in understanding the following idea:
Suppose we have a function $f(x)$ such that $f \in L^1([0,\infty)), |f(x)| \leq C \exp(-\rho x)$, $\int_0^\infty f > 0$. Further, suppose $f$ oscillates $n$ times over the $x$ axis in the sense that $\exists$ constants $M_1<M_2<...<M_n$ such that $f(x) \geq 0 $ on $[0,M_1] \cup [M_2,M_3] \cup ... \cup [M_{n-1},M_n]$ and $f(x) \leq 0$ on $[M_1,M_2] \cup [M_3,M_4] \cup ... \cup [M_n,\infty)$.
Are there any generalizations as to how the Laplace transform $\int_0^\infty e^{-sx} f(x) \, \mathrm{d}x$ will behave? In particular, if $s$ is positive real, can we say anything about the Laplace transform? Are there any general ways to understand the graph of the laplace transform for positive $s$?
I realize that $f$ is very general, which is exactly why I don't expect a definitive answer. I also realize there are results if $f$ is periodic. I do not necessarily wish my $f$ to be periodic. I'm mostly curious if anyone has direction for understanding this idea and the conditions on $f$.
Are there any good books or papers you recommend? Thank you for your help.
