In this work we study the harmonic analysis in the gyrogroup of proper velocities (PV) using the gyro-language of the analytic hyperbolic geometry developed by A.A. Ungar. PV addition is the relativistic addition of proper velocities in special relativity and it is related with the hyperboloid model of hyperbolic geometry. We study the generalized translation operator, the convolution operator, the generalized Laplace-Beltrami operator and its eigenfunctions, the Poisson transform, the generalized Helgason-Fourier transform, its inverse, and Plancherel’s theorem. At the limit when the radius of the hyperboloid tends to infinity the generalized harmonic analysis on the hyperboloid becomes the harmonic Euclidean analysis in Rn.

Author: Ferreira M

Published in: Banach Journal of Mathematical Analysis