Parseval like theorem for Laplace transform?

Question

I was wondering if there any parseval like theorem for Laplce transform? Some statement such as

$\int f(x)g(x) dx=\int F(s)G(s) ds$

where $F(s)=\mathcal{L}\{f(t)\}$ and $G(s)=\mathcal{L}\{g(t)\}$?

Answer 1

The Laplace transform doesn't have the same sort of symmetry as the Fourier transform; the Fourier transform and the inverse Fourier transform are essentially the same thing, not so for the Laplace transform. Besides which, many Laplace transformable functions are not integrable. For example, any polynomial has a Laplace transform. This is one of the main "problems" with the Laplace transform: its domain is so large that it's really tricky to characterize the image. For example, the Fourier transform maps $L^p$ to $L^q$ (where $p,q$ are conjugate exponents) or Schwarz class to Schwarz class. No such statements hold for the Laplace transform.

Answer 2

The Laplace transform of an exponentially-bounded function is essentially the Fourier transform at complex "frequencies". There is a Parseval theorem, but it involves integration on a vertical line in the $s$-plane. See e..g this MathOverflow question and answer.