There are a number of variations on the Laplace transform that turn up all over math. Some examples:
- $\int_{-\infty}^{\infty} f(t)e^{-st} dt$ - The Laplace transform
- $\sum_{-\infty}^{\infty} f(t)z^{-t}$ - The Z-transform
- $\int_{-\infty}^{\infty} f(t)e^{-i\omega t} dt$ - The Fourier transform
- $\sum_{-\infty}^{\infty} f(t)e^{-i\omega t}$ - The discrete-time Fourier transform
The summations are over t. There are some tweaks to make regarding normalization and etc, but this is the basic gist of it.
The Mellin transform is just as important, and also comes with several obvious variations. However, the Mellin "universe" seems a bit less organized than the Laplace universe, and names for these variations are harder for me to find.
Here's what I have so far:
- $\int_{0}^{\infty} f(t)t^{s-1} dt$ - The Mellin transform
- $\sum_{-\infty}^{\infty} f(n)n^{-s}$ - The Dirichlet transform
- $\int_{0}^{\infty} f(t)t^{-i\omega} dt$ - ????
- $\sum_{0}^{\infty} f(t)t^{-i\omega}$ - ????
Again, there are a few minor tweaks to be made, some of which are especially annoying due to that $s-1$ exponent on the Mellin transform vs the $s$ exponent on the Dirichlet transform, but you get the gist.
Do any of these transforms have names, and are they ever commonly used? The last two, which fit the pattern Laplace:Fourier::Mellin:________, seem to me like they'd be particularly useful in digital signal processing, but I can't find any reference to them anywhere.
