\[\renewcommand{\vv}[1]{\boldsymbol{#1}} \renewcommand{\mm}[1]{\mathrm{#1}} \renewcommand{\mmm}[1]{\mathrm{#1}} \renewcommand{\cc}[1]{\mathcal{#1}} \renewcommand{\ff}[1]{\mathfrak{#1}} \renewcommand{\oo}[1]{\operatorname{#1}} \renewcommand{\cc}[1]{\mathcal{#1}}\]

“An integral transform with scale invariance like the Fourier transform has shift invariance”. From my perspective, useful for analysing multiplicative products of random variables, their reciprocal and powers, much as the Fourier transform is useful for sums of scaled random variables.

For now, see the Mellin transform on Wikipedia.

Approachable references seem to be Bertrand, Bertrand, and Ovarlez (2000); Flajolet, Gourdon, and Dumas (1995); Galambos and Simonelli (2004).

## References

Adams, David R., and Lars I. Hedberg. 1999.

*Function Spaces and Potential Theory*. Springer Science & Business Media.
Bertrand, Jacqueline, Pierre Bertrand, and Jean-Philippe Ovarlez. 2000. “The Mellin Transform.” In

*The Transforms and Applications Handbook*, edited by Alexander D. Poularikas, 2nd ed. The Electrical Engineering Handbook Series. Boca Raton, Fla: CRC Press.
Brychkov, I︠U︡ A., O. I. Marichev, and Nikolay V. Savischenko. 2019.

*Handbook of Mellin Tranforms*. Advances in Applied Mathematics. Boca Raton: CRC Press, Taylor & Francis Group.
Cohen, L. 1993. “The Scale Representation.”

*IEEE Transactions on Signal Processing*41 (12): 3275–92.
Davies, Brian. 2002.

*Integral Transforms and Their Applications*. 3rd edition. New York: Springer.
De Sena, Antonio, and Davide Rocchesso. 2007. “A Fast Mellin and Scale Transform.”

*EURASIP J. Appl. Signal Process.*2007 (1): 75–75.
Debnath, Lokenath, and Dambaru Bhatta. 2014.

*Integral Transforms and Their Applications*. 3rd edition. Chapman and Hall/CRC.
Dufresne, Daniel. 1998. “Algebraic Properties of Beta and Gamma Distributions, and Applications.”

*Advances in Applied Mathematics*20 (3): 285–99.
Flajolet, Philippe, Xavier Gourdon, and Philippe Dumas. 1995. “Mellin Transforms and Asymptotics: Harmonic Sums.”

*Theoretical Computer Science*144 (1): 3–58.
Galambos, János, and Italo Simonelli. 2004.

*Products of random variables : applications to problems of physics and to arithmetical functions*. New York : Marcel Dekker.
Hackmann, Daniel, and Alexey Kuznetsov. 2016. “Approximating Lévy Processes with Completely Monotone Jumps.”

*The Annals of Applied Probability*26 (1): 328–59.
Jánossy, L., and H. Messel. 1950. “Fluctuations of the Electron-Photon Cascade - Moments of the Distribution.”

*Proceedings of the Physical Society. Section A*63 (10): 1101.
Lee, Juho, Xenia Miscouridou, and François Caron. 2019. “A Unified Construction for Series Representations and Finite Approximations of Completely Random Measures.”

*arXiv:1905.10733 [Cs, Math, Stat]*, May.
Marchand, Ugo, and Geoffroy Peeters. 2014. “The Modulation Scale Spectrum And Its Application To Rhythm-Content Analysis.” In

*DAFX (Digital Audio Effects)*. Erlangen, Germany.
Nuzman, Carl J., and H. Vincent Poor. 2000. “Linear Estimation of Self-Similar Processes via Lamperti’s Transformation.”

*Journal of Applied Probability*37 (2): 429–52.
Polyanin, A. D., and A. V. Manzhirov. 1998.

*Handbook of Integral Equations*. Boca Raton, Fla: CRC Press.
Poularikas, Alexander D., ed. 2000.

*The Transforms and Applications Handbook*. 2nd ed. The Electrical Engineering Handbook Series. Boca Raton, Fla: CRC Press.
Schiff, Joel L. 1999.

*The Laplace Transform: Theory and Applications*. 1999th edition. New York: Springer.
Sena, Antonio De, and Davide Rocchesso. 2004. “A Fast Mellin Transform with Applications in DAFX,” 5.

Simon, Barry. 2015.

*Real Analysis*. A Comprehensive Course in Analysis 1.0. UNIVERSITIES PRESS.
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