Physics Forums

Physics Forums (http://www.physicsforums.com/index.php)
-   Linear & Abstract Algebra (http://www.physicsforums.com/forumdisplay.php?f=75)
-   -   Laplace transform as rotation. In what space? (http://www.physicsforums.com/showthread.php?t=584191)

mclaudt Mar5-12 08:21 PM

Laplace transform as rotation. In what space?
 
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?

mclaudt Mar5-12 08:36 PM

Re: Laplace transform as rotation. In what space?
 
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.

mclaudt Mar5-12 09:56 PM

Re: Laplace transform as rotation. In what space?
 
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.

mclaudt Mar17-12 08:25 PM

Re: Laplace transform as rotation. In what space?
 
Here some relevant links but none of them solve the problem

http://mathoverflow.net/questions/38...efinition/2141
http://www.physicsforums.com/showthread.php?t=155709
http://mathoverflow.net/questions/28...ral-transforms
http://math.stackexchange.com/questi...ns-for-dummies
http://forums.xkcd.com/viewtopic.php?f=17&t=62999
http://www.reddit.com/r/math/comment...lace_transform


All times are GMT -5. The time now is 10:11 AM.

Powered by vBulletin Copyright ©2000 - 2014, Jelsoft Enterprises Ltd.
© 2014 Physics Forums