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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.

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I know that the discrete Mellin transform was defined by V.S.Ryko: http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=ivm&paperid=5138&option_lang=rus English reference: Soviet Mathematics (Izvestiya VUZ. Matematika), 1991, 35:8, 63–66
He also developed a very strong method with many page tables to sum series based on it (reference [4] in the cited paper). Formula (1) in his paper is an integral representation for any Derichlet series also. Unfortunately all that is fully forgotten now and sometimes is rediscovered by and by...

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    $\begingroup$ Note also a chapter 11 in the book Poularikas, A. D. (Ed.). The Transforms and Applications Handbook. Boca Raton, FL: CRC Press, 1995. It contains a section on Discreet Mellin Tr. $\endgroup$ – Sergei May 11 '15 at 15:28

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