1) A Fourier Transform basically tells us that any LIMITED signal can be converted into an infinite sum of sinosoids of diferent amplitudes and frequencies:

2) A Transfer Function using FT tells us 2 things: the amplitude of each e^jwt term and the phase for each.

3) Since the Fourier Transform only works for limited signals (in amplitude and time) we use the Laplace Transform for ALL signals, which is basicaly the same as a FT, but now s=σ+jω. In other words, it tells us that any signal can be written as a sum of sinosoids TIMES an exponencial term (e^σt)

http://upload.wikimedia.org/math/3/6/1/36193c7b97db891595cec12dd627b814.png

4) A LT transfer function is different: it tells us how the amplitude of each term e^st changes depending on the σ we choose to use. In other words, we consider only the first term of the sum, in which ω=0, and thus e^s=e^σ and analyse how the magnitude of this first term changes as we change σ

I am very sorry if I was hard to understand, but right now this is how I understand these transforms.

Is this wrong or not?

Thank you