Suppose I have two functions $f,g:\mathbb{R}\to[0,\infty)$ (in fact they are nonnegative measures), and I would like to find the integral of their product, $\int_{-\infty}^\infty f(x)g(x)dx$.
However, I only know the two-sided Laplace transforms of the two functions. That is, I have equations for $$ F(s) = \int_{-\infty}^\infty e^{sx}f(x)dx,\qquad\text{and}\qquad G(t) = \int_{-\infty}^\infty e^{tx}g(x)dx. $$ Is there a 'nice' formula for the integral $\int_{-\infty}^\infty f(x)g(x)dx$ in terms of the functions $F$ and $G$, without first having to perform their inverse Laplace transforms?
Since the Laplace-transform is just the Fourier-transform rotated by 90 degrees, one can do the usual tricks: (Up to some constants) The integral is the Fourier-Laplace-transform evaluated at zero and the F/L-transform of a product is the convolution of the individual transforms. So we expect a formula of the form $\int f(x)g(x) = const\cdot (F\ast G)(0)$.
More precisely: If $\widehat{\cdot}$ denotes the unitary Fourier transform $\widehat{f}(\xi) = (2\pi)^{-n/2}\int_{\mathbb{R}^n} f(x)e^{-ix\xi} dx$ (if $f\in L^1$), then the constant works out to be 1: $$\int_\mathbb{R} f(x) g(x) dx = \sqrt{2\pi}\widehat{f\cdot g}(0) = (\widehat{f} \ast \widehat{g})(0) = (F(is)\ast G(is))(0) = \int_\mathbb{R} F(is)G(-is)ds$$ (assuming of course that $f,g,F$ and $G$ are sufficiently nice)