“An integral transform with scale invariance like the Fourier transform has shift invariance.” From my perspective, it’s useful for analysing multiplicative products of random variables, their reciprocals 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).
1 Incoming
2 References
Adams, and Hedberg. 1999. Function Spaces and Potential Theory.
Bertrand, Bertrand, and Ovarlez. 2000. “The Mellin Transform.” In The Transforms and Applications Handbook. The Electrical Engineering Handbook Series.
Brychkov, Marichev, and Savischenko. 2019. Handbook of Mellin Tranforms. Advances in Applied Mathematics.
Cohen. 1993. “The Scale Representation.” IEEE Transactions on Signal Processing.
Davies. 2002. Integral Transforms and Their Applications.
De Sena, and Rocchesso. 2007. “A Fast Mellin and Scale Transform.” EURASIP J. Appl. Signal Process.
Debnath, and Bhatta. 2014. Integral Transforms and Their Applications.
Dufresne. 1998. “Algebraic Properties of Beta and Gamma Distributions, and Applications.” Advances in Applied Mathematics.
Flajolet, Gourdon, and Dumas. 1995. “Mellin Transforms and Asymptotics: Harmonic Sums.” Theoretical Computer Science.
Galambos, and Simonelli. 2004. Products of random variables : applications to problems of physics and to arithmetical functions.
Hackmann, and Kuznetsov. 2016. “Approximating Lévy Processes with Completely Monotone Jumps.” The Annals of Applied Probability.
Jánossy, and Messel. 1950. “Fluctuations of the Electron-Photon Cascade - Moments of the Distribution.” Proceedings of the Physical Society. Section A.
Lee, Miscouridou, and Caron. 2019. “A Unified Construction for Series Representations and Finite Approximations of Completely Random Measures.” arXiv:1905.10733 [Cs, Math, Stat].
Marchand, and Peeters. 2014. “The Modulation Scale Spectrum And Its Application To Rhythm-Content Analysis.” In DAFX (Digital Audio Effects).
Nuzman, and Poor. 2000. “Linear Estimation of Self-Similar Processes via Lamperti’s Transformation.” Journal of Applied Probability.
Polyanin, and Manzhirov. 1998. Handbook of Integral Equations.
Poularikas, ed. 2000. The Transforms and Applications Handbook. The Electrical Engineering Handbook Series.
Schiff. 1999. The Laplace Transform: Theory and Applications.
Sena, and Rocchesso. 2004. “A Fast Mellin Transform with Applications in DAFX.”
Simon. 2015. Real Analysis. A Comprehensive Course in Analysis 1.0.
Zayed, ed. 2019. Handbook of Function and Generalized Function Transformations.