Laplace transform as rotation. In what space?

[mclaudt]

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?

 

[mclaudt]

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]

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.

 

[mclaudt]

Here some relevant links but none of them solve the problem

http://mathoverflow.net/questions/383/motivating-the-laplace-transform-definition/2141
https://www.physicsforums.com/showthread.php?t=155709
http://mathoverflow.net/questions/2809/intuition-for-integral-transforms
http://math.stackexchange.com/questions/6661/laplace-transformations-for-dummies
http://forums.xkcd.com/viewtopic.php?f=17&t=62999
http://www.reddit.com/r/math/comments/eew4z/intuition_behind_the_laplace_transform

 

[blue_raver22]

I understand your desire to grasp the main idea of the Laplace transform without getting bogged down in complicated mathematical terminology. So, to put it simply, the Laplace transform can indeed be thought of as a rotation of basis in a certain space. However, this space is not the same as the one we typically think of in terms of physical space, but rather a mathematical space of functions.

To understand this, let's first look at the Fourier transform. This transform takes a function in the time domain and converts it into a function in the frequency domain. In other words, it allows us to see how much of each frequency component makes up a given function. This can be thought of as a rotation of basis, because we are essentially changing the way we represent the function.

Now, the Laplace transform is a generalization of the Fourier transform to complex frequencies. This means that instead of just considering real-valued frequencies, we also consider complex frequencies. In this case, the "space" we are working in is the space of complex-valued functions of complex argument. So, the basis we are rotating from is the basis of delta-functions (which represent a single frequency) to a new basis of functions of the form e^{(\alpha + i\beta) t}, where alpha and beta are real numbers representing the real and imaginary parts of the complex frequency.

As for the limits of summation in the Laplace transform, this has to do with the convergence of the integral. The Laplace transform is defined as an integral over a specific range of complex frequencies. In the direct transform, this range is from 0 to infinity, which corresponds to the line (-\infty, +\infty) in the complex plane. In the reverse transform, the range is from -\infty to +\infty, which corresponds to the line (\gamma - i\infty, \gamma + i\infty) in the complex plane. These ranges are chosen based on the properties of the Laplace transform and ensure that the integral converges.

Overall, the idea of the Laplace transform as a rotation of basis in a mathematical space is a useful way to understand its purpose and how it relates to the Fourier transform. However, it is important to also understand the mathematical definitions and properties in order to fully grasp its applications and implications in physics and other fields.