Sorry for thumb terminology, I just would like to grasp the main idea, as a physicist, without unnecessary complications, associated with system of axioms and definitions. Fourier transform can be seen as rotation of basis in space of all complex-valued functions from basis of delta-functions to a new basis of waves [itex]e^{i\omega t}[/itex]. Laplace transform can be seen as generalization of Fourier transform to complex frequencies. Is it correct to see the Laplace transform as a rotation of basis in space of complex-valued functions of complex argument, from delta-functions basis to a new basis of [itex]e^{(\alpha + i\beta) t}[/itex]and, if it is so, what is the basis in that space, and why does summation go only along line [itex](-\infty, +\infty)[/itex] in direct transform and [itex](\gamma - i\infty, \gamma + i\infty)[/itex] in reverse transform?
There is also an operation known as Wick rotation, it is a candidate to interpret the interconnection between Fourier and Laplace, it could be interesting if they both constitute some beautiful scheme.
It seems that the key point is the analytical continuation. The raw form of Laplace transform is [itex]F(s) = \int_{ℂ}e^{-su}f(u)du[/itex], where [itex]f(u)[/itex] is analytical continuation of [itex]f(t)[/itex] to complex plane. And in that case the Laplace transform will be the rotation of basis in [itex]{f(u)}[/itex] space of complex functions of complex argument, just as Fourier transform is the rotation of basis in [itex]{f(t)}[/itex] space of complex functions of real argument, and integral is in both cases just a generalization of matrix multiplication to infinite dimension. It's just guess, I'm not sure if it is correct to define [itex]F(s)[/itex] that way.
Here some relevant links but none of them solve the problem http://mathoverflow.net/questions/383/motivating-the-laplace-transform-definition/2141 https://www.physicsforums.com/showthread.php?t=155709 http://mathoverflow.net/questions/2809/intuition-for-integral-transforms http://math.stackexchange.com/questions/6661/laplace-transformations-for-dummies http://forums.xkcd.com/viewtopic.php?f=17&t=62999 http://www.reddit.com/r/math/comments/eew4z/intuition_behind_the_laplace_transform