I tend to think of the Fourier Transform (FT) as projecting a function onto a basis of cosines and sines. The Laplace Transform (LT) has a similar form to the FT, except it has been generalised.
Since the FT projects onto a basis of cosines and sines, cosines and sines turn up as dirac-deltas---this makes sense to me. Now, I was wondering upon what basis the LT projects a function. I figure that I could get a hint about what the basis upon which the LT projects, by looking at the ILT of a dirac delta. $$ \mathcal{F}^{-1}\{\delta(\omega+a_0)\} \propto e^{j a_0 t} \\ \mathcal{L}^{-1}\{\delta(s+a_0)\} =\ ? $$ So the IFT of a dirac delta is making a purely complex exponential (makes sense) but I've no idea what the ILT would do.
Currently, I'm just viewing the LT as basically a FT, except that it multiples the function being transformed by some real exponential and then does a FT.
Could anyone perhaps shed some light on whether the LT is actually projecting onto a basis (like the FT is) and what that basis is?
Thanks.