Laplace transform as rotation. In what space?

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Discussion Overview

The discussion revolves around the conceptual understanding of the Laplace transform, particularly its relationship to the Fourier transform and its interpretation as a rotation in function space. Participants explore the mathematical and theoretical implications of these transforms, including their bases and the nature of integration in complex spaces.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant suggests that the Laplace transform can be viewed as a generalization of the Fourier transform to complex frequencies, questioning the nature of the basis in this context.
  • Another participant introduces the concept of Wick rotation as a potential link between the Fourier and Laplace transforms, proposing that they may form an interconnected scheme.
  • A different participant emphasizes the role of analytical continuation in defining the Laplace transform, comparing it to the Fourier transform and suggesting that both involve rotations in their respective function spaces.
  • Some participants express uncertainty about the definitions and interpretations being proposed, indicating that these ideas are still under exploration.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the interpretations and definitions surrounding the Laplace transform and its relationship to the Fourier transform. Multiple competing views and uncertainties remain present in the discussion.

Contextual Notes

Participants note limitations in their understanding, particularly regarding the definitions of bases in the relevant function spaces and the implications of integration paths in the complex plane.

Who May Find This Useful

This discussion may be of interest to those exploring advanced mathematical concepts in transforms, particularly in the context of physics and engineering applications.

mclaudt
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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?
 
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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 F(s) = \int_{ℂ}e^{-su}f(u)du, where f(u) is analytical continuation of f(t) to complex plane. And in that case the Laplace transform will be the rotation of basis in {f(u)} space of complex functions of complex argument, just as Fourier transform is the rotation of basis in {f(t)} 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 F(s) that way.
 

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