Gyroharmonic Analysis on Relativistic Gyrogroups

Document Type : Special Issue: AIMC 51


Center for Research and Development in Mathematics and Applications (CIDMA), University of Aveiro 3810-193 Aveiro, Portugal


‎Einstein‎, ‎Möbius‎, ‎and Proper Velocity gyrogroups are relativistic gyrogroups that appear as three different realizations of the proper Lorentz group in the real Minkowski space-time Rn,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⇒‎ +∞, the resulting gyroharmonic analysis tends to the standard Euclidean harmonic analysis on Rn, thus unifying hyperbolic and Euclidean harmonic analysis‎.


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