Abstract :
Einstein, M quot;{o}bius, and Proper Velocity gyrogroups are relativistic gyrogroups that appear as three different realizations of the proper Lorentz group in the real Minkowski spacetime $bkR^{n,1}.$ Using the gyrolanguage we study their gyroharmonic analysis. Although there is an algebraic gyroisomorphism between the three models we show that there are some differences between them. Our study focus on the translation and convolution operators, eigenfunctions of the LaplaceBeltrami operator, Poisson transform, FourierHelgason transform, its inverse, and Plancherel’s Theorem. We show that in the limit of large $t,$ $t rightarrow +infty,$ the resulting gyroharmonic analysis tends to the standard Euclidean harmonic analysis on ${mathbb R}^n,$ thus unifying hyperbolic and Euclidean harmonic analysis.
