Gyroharmonic Analysis on Relativistic Gyrogroups

‎Einstein‎, ‎M"{o}bius‎, ‎and Proper Velocity gyrogroups are relativistic gyrogroups that appear as three different realizations of the proper Lorentz group in the real Minkowski space-time $bkR^{n,1}.$ 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 Laplace-Beltrami operator‎, ‎Poisson transform‎, ‎Fourier-Helgason 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.