Gyroharmonic Analysis on Relativistic Gyrogroups

Document Type: Special Issue: International Conference on Architecture and Mathematics

Author

Polytechnic Institute of Leiria, Portugal

Abstract

‎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.

Keywords

Main Subjects


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