Banach Journal of Mathematical Analysis

Harmonic analysis on the proper velocity gyrogroup

Milton Ferreira

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In this article we study harmonic analysis on the proper velocity (PV) gyrogroup using the gyrolanguage of analytic hyperbolic geometry. This PV addition is the relativistic addition of proper velocities in special relativity, and it is related with the hyperboloid model of hyperbolic geometry. The generalized harmonic analysis depends on a complex parameter z and on the radius t of the hyperboloid, and it comprises the study of the generalized translation operator, the associated convolution operator, the generalized Laplace–Beltrami operator and its eigenfunctions, the generalized Poisson transform and its inverse, the generalized Helgason–Fourier transform and its inverse, and Plancherel’s theorem. In the limit of large t, t+, the generalized harmonic analysis on the hyperboloid tends to the standard Euclidean harmonic analysis on Rn, thus unifying hyperbolic and Euclidean harmonic analysis.

Article information

Banach J. Math. Anal., Volume 11, Number 1 (2017), 21-49.

Received: 30 November 2015
Accepted: 15 February 2016
First available in Project Euclid: 19 October 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 43A85: Analysis on homogeneous spaces
Secondary: 43A30: Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc. 43A90: Spherical functions [See also 22E45, 22E46, 33C55] 44A35: Convolution 20N05: Loops, quasigroups [See also 05Bxx]

PV gyrogroup Laplace–Beltrami operator eigenfunctions generalized Helgason–Fourier transform Plancherel’s theorem


Ferreira, Milton. Harmonic analysis on the proper velocity gyrogroup. Banach J. Math. Anal. 11 (2017), no. 1, 21--49. doi:10.1215/17358787-3721232.

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