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.