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I've never understood the use of the Laplace transform. Perhaps that's due to my mathematical exposure (theoretical qualitative analysis of pdes). Since the Lap

by dls2016 6y ago

I've never understood the use of the Laplace transform. Perhaps that's due to my mathematical exposure (theoretical qualitative analysis of pdes). Since the Laplace transform lacks the duality of the Fourier transform, it doesn't seem to have a place in research mathematics. But I probably think of a dozen fundamental uses of the Fourier transform, from Bourgain spaces to evaluating oscillatory integrals. And if you're working on some manifold with curvature then you generally need to be familiar with the eigenfunctions of the Laplacian on that manifold... not the basis of the Laplace transform. I also know a bit of signal processing\numerical analysis, and I'm not familiar with any practical uses of the Laplace transform there. I don't believe it's used in the numerical solution of pdes or odes, whereas spectral methods are a huge area of study and (until recently, I think) were used in the GFS weather model. And most time series analysis tools either apply the Fourier transform or bail out of this approach and use statistical tools. My version of Greenspun's 10th rule goes: any sufficiently complex program includes an FFT. Can anyone help me out here? Is there a problem/theorem the Laplace transform solves/proves which the Fourier transform doesn't?

chessweb01 6y ago
https://duckduckgo.com/?t=ffab&q=applications+laplace+transform&atb=v100-1&ia=web https://duckduckgo.com/?t=ffab&q=applications+laplace+transf...
- dls2016 6y ago
  har har But seriously, do you have a favorite example where the Laplace transform is used to prove a theorem or used in practice to solve a problem? I'm familiar with the undergraduate differential equations examples. But there are plenty of things taught at the undergraduate level which are tractable and helpful to build intuition but either a) aren't important from a research perspective or b) aren't used in practice. The Fourier transform has both.
  - SAI_Peregrinus 6y ago
    All the time in AC circuits. Especially for anything RF-related. It's vastly, vastly easier to work in the (complex) frequency domain. Antenna and filter design are pretty much all done in the s-domain.
    - dls2016 6y ago
      So is s-domain superior to Fourier domain for stability analysis only, or are there other reasons? (I know enough signal processing to be dangerous.)
whatshisface 6y ago
The Fourier transform is a line cut out of the Laplace transform, and the Laplace transform is the analytic continuation of the Fourier transform. So you should not be surprised to see the Fourier transform show up in all applications, because there is an FFT but no FLT.
teleforce 6y ago
For a good explanation on Laplace Transform please check the video presentation link that I've provided in my other comments. As for the Laplace Transform, it is mainly use in control system applications where the input/output include transient/damping/forcing signal waveforms (on and off unit circle) not only clean steady state signal waveforms (on unit circle). This paper provides a good overview of the sample usages of a Laplace Transform in Electrical and Electronics Engineering [1]. If what you meant by the duality Fourier Transform as FFT/IFFT, Laplace Transform has the equivalent in the form of Chirp-Z Transform (CZT) and recently discovered inverse CZT (ICZT), and the original HN discussions link of the discovery is also provided in my other comments. For CZT/ICZT potential useful application please check the other/older HN topic comments in [2]. Perhaps we should just wait and watch for the torrent of patent filings on this CZT/ICZT topic if the claim of ICZT is really true and feasible. [1]http://sces.phys.utk.edu/~moreo/mm08/sarina.pdf http://sces.phys.utk.edu/~moreo/mm08/sarina.pdf [2]https://news.ycombinator.com/item?id=21232296 https://news.ycombinator.com/item?id=21232296
- dls2016 6y ago
  I see your point about stability analysis. I will take a look into CZT, I recall the HN post at the time but didn't look into it much.
mgraczyk 6y ago
Remember in linear algebra how you spent most of the time learning about eigenvalues and eigenvectors, and particularly how to "diagonalize" a matrix `A` into `A=PDP^-1`? Doing this makes `A` easier to work with, so problems that include `A` are often easier to solve if you replace `A` with `PDP^-1`. The laplace transform is the same thing, but instead of matrices, it works on derivatives. Equations involving `d/dt` are often made easier to work with by instead using `s`. Longer answer: https://www.quora.com/What-is-the-purpose-of-Laplace-transforms-in-controls-theory https://www.quora.com/What-is-the-purpose-of-Laplace-transfo...
- dls2016 6y ago
  This is true of the Fourier transform, too. The duality properties of the FT make it a nice theoretical tool. And the existence of the FFT make it a nice practical tool, too.
qppo 6y ago
> Can anyone help me out here? Is there a problem/theorem the Laplace transform solves/proves which the Fourier transform doesn't? BIBO stability of IIR systems
- dls2016 6y ago
  Interesting, but I'm not seeing what this has to do with Fourier or Laplace. Mathematically, the statement is: if x is L^{\infty} and h is L^1, then y=x*h is L^{\infty}. The proof of this is very routine... just write down the definition of the convolution and stare at it. Now it's true that the Fourier transform is not well-behaved on L^{\infty}. Any book on harmonic analysis will discuss this point in great lengths. But the context in which these questions are discussed (singular integral operators on spaces like BMO, which contains L^{\infty}) doesn't tend to include the Laplace transform in a useful manner. Someday I'll have to properly learn harmonic analysis to sort out these questions for myself.

chessweb01 6y ago

https://duckduckgo.com/?t=ffab&q=applications+laplace+transform&atb=v100-1&ia=web https://duckduckgo.com/?t=ffab&q=applications+laplace+transf...

dls2016 6y ago
har har But seriously, do you have a favorite example where the Laplace transform is used to prove a theorem or used in practice to solve a problem? I'm familiar with the undergraduate differential equations examples. But there are plenty of things taught at the undergraduate level which are tractable and helpful to build intuition but either a) aren't important from a research perspective or b) aren't used in practice. The Fourier transform has both.
- SAI_Peregrinus 6y ago
  All the time in AC circuits. Especially for anything RF-related. It's vastly, vastly easier to work in the (complex) frequency domain. Antenna and filter design are pretty much all done in the s-domain.
  - dls2016 6y ago
    So is s-domain superior to Fourier domain for stability analysis only, or are there other reasons? (I know enough signal processing to be dangerous.)

whatshisface 6y ago

The Fourier transform is a line cut out of the Laplace transform, and the Laplace transform is the analytic continuation of the Fourier transform. So you should not be surprised to see the Fourier transform show up in all applications, because there is an FFT but no FLT.

teleforce 6y ago

For a good explanation on Laplace Transform please check the video presentation link that I've provided in my other comments. As for the Laplace Transform, it is mainly use in control system applications where the input/output include transient/damping/forcing signal waveforms (on and off unit circle) not only clean steady state signal waveforms (on unit circle). This paper provides a good overview of the sample usages of a Laplace Transform in Electrical and Electronics Engineering [1]. If what you meant by the duality Fourier Transform as FFT/IFFT, Laplace Transform has the equivalent in the form of Chirp-Z Transform (CZT) and recently discovered inverse CZT (ICZT), and the original HN discussions link of the discovery is also provided in my other comments. For CZT/ICZT potential useful application please check the other/older HN topic comments in [2]. Perhaps we should just wait and watch for the torrent of patent filings on this CZT/ICZT topic if the claim of ICZT is really true and feasible. [1]http://sces.phys.utk.edu/~moreo/mm08/sarina.pdf http://sces.phys.utk.edu/~moreo/mm08/sarina.pdf [2]https://news.ycombinator.com/item?id=21232296 https://news.ycombinator.com/item?id=21232296

dls2016 6y ago
I see your point about stability analysis. I will take a look into CZT, I recall the HN post at the time but didn't look into it much.

mgraczyk 6y ago

Remember in linear algebra how you spent most of the time learning about eigenvalues and eigenvectors, and particularly how to "diagonalize" a matrix `A` into `A=PDP^-1`? Doing this makes `A` easier to work with, so problems that include `A` are often easier to solve if you replace `A` with `PDP^-1`. The laplace transform is the same thing, but instead of matrices, it works on derivatives. Equations involving `d/dt` are often made easier to work with by instead using `s`. Longer answer: https://www.quora.com/What-is-the-purpose-of-Laplace-transforms-in-controls-theory https://www.quora.com/What-is-the-purpose-of-Laplace-transfo...

dls2016 6y ago
This is true of the Fourier transform, too. The duality properties of the FT make it a nice theoretical tool. And the existence of the FFT make it a nice practical tool, too.

qppo 6y ago

> Can anyone help me out here? Is there a problem/theorem the Laplace transform solves/proves which the Fourier transform doesn't? BIBO stability of IIR systems

dls2016 6y ago
Interesting, but I'm not seeing what this has to do with Fourier or Laplace. Mathematically, the statement is: if x is L^{\infty} and h is L^1, then y=x*h is L^{\infty}. The proof of this is very routine... just write down the definition of the convolution and stare at it. Now it's true that the Fourier transform is not well-behaved on L^{\infty}. Any book on harmonic analysis will discuss this point in great lengths. But the context in which these questions are discussed (singular integral operators on spaces like BMO, which contains L^{\infty}) doesn't tend to include the Laplace transform in a useful manner. Someday I'll have to properly learn harmonic analysis to sort out these questions for myself.