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.
I will take a look into CZT, I recall the HN post at the time but didn't look into it much.
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-transfo...
BIBO stability of IIR systems
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.
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.
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?