Let's try to find a function $\psi(x)$ such that for Laplace transform $\tilde{f}(p)=\int_0^{\infty} f(y) e^{-py} dy$ one has $f(x)=\int_0^{\infty} \tilde{f}(p)\psi(px)dp$ (here we do not specify classes of functions, for which this should hold).

In other words, $\tilde{\psi}(p)=\delta_1(p)$ in the sense of distributions.

Or, Stiltjes transform $\int_0^{\infty} \frac{\psi(t)}{t+y} dt=e^{-y}, y>0$.

(just represent $\frac1{t+y}=\int_0^{\infty} e^{-q(t+y)}dq$ and change order of integration).

So the question(s) is(are):

Does such function exist, if it exists, what are its properties, where is it written about all this stuff and so on.