Laplace transform as rotation. In what space?

  1. Mar 5, 2012 #1
    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?
     
  2. jcsd
  3. Mar 5, 2012 #2
    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.
     
  4. Mar 5, 2012 #3
    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.
     
  5. Mar 17, 2012 #4
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