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January 2017 Harmonic analysis on the proper velocity gyrogroup
Milton Ferreira
Banach J. Math. Anal. 11(1): 21-49 (January 2017). DOI: 10.1215/17358787-3721232


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.


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Milton Ferreira. "Harmonic analysis on the proper velocity gyrogroup." Banach J. Math. Anal. 11 (1) 21 - 49, January 2017.


Received: 30 November 2015; Accepted: 15 February 2016; Published: January 2017
First available in Project Euclid: 19 October 2016

zbMATH: 1354.43006
MathSciNet: MR3562359
Digital Object Identifier: 10.1215/17358787-3721232

Primary: 43A85
Secondary: 20N05 , 43A30 , 43A90 , 44A35

Keywords: Eigenfunctions , generalized Helgason–Fourier transform , Laplace–Beltrami operator , Plancherel’s theorem , PV gyrogroup

Rights: Copyright © 2017 Tusi Mathematical Research Group

Vol.11 • No. 1 • January 2017
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