- #1
mclaudt
- 6
- 0
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 e^{i\omega t}.
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 e^{(\alpha + i\beta) t}and, if it is so, what is the basis in that space, and why does summation go only along line (-\infty, +\infty) in direct transform and (\gamma - i\infty, \gamma + i\infty) in reverse transform?
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 e^{i\omega t}.
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 e^{(\alpha + i\beta) t}and, if it is so, what is the basis in that space, and why does summation go only along line (-\infty, +\infty) in direct transform and (\gamma - i\infty, \gamma + i\infty) in reverse transform?