A Laplace Transform is based on the integral:
$F(\xi) = \int_0^{\infty} f(x) e^ {-\xi x}\,dx.$
In a roundabout way, a Fourier transform can get to $\hat{f}(\xi) = \int_{-\infty}^{\infty} f(x)\ e^{- 2\pi i x \xi}\,dx,$
Also, they both seem to use convolutions and transposes in "indirect" forms of "multiplication."