2014 Harmonic Analysis on the Einstein Gyrogroup
Milton Ferreira
J. Geom. Symmetry Phys. 35: 21-60 (2014). DOI: 10.7546/jgsp-35-2014-21-60

## Abstract

In this paper we study harmonic analysis on the Einstein gyrogroup of the open ball of ${\mathbb R}^n, n \in \mathbb{N},$ centered at the origin and with arbitrary radius $t \in \mathbb{R}^+,$ associated to the generalised Laplace-Beltrami operator $$L_{\sigma,t} = \left( 1 - \frac{\|x\|^2}{t^2} \right) \!\left( \Delta - \sum_{i,j=1}^n \frac{x_i x_j}{t^2} \frac{\partial^2}{\partial x_i \partial x_j} - \frac{\kappa}{t^2} \sum_{i=1}^n x_i \frac{\partial}{\partial x_i} + \frac{\kappa(2-\kappa)}{4t^2} \right)$$ where $\kappa=n+\sigma$ and $\sigma \in {\mathbb R}$ is an arbitrary parameter. The generalised harmonic analysis for $L_{\sigma,t}$ gives rise to the $(\sigma,t)$-translation, the $(\sigma,t)$- convolution, the $(\sigma,t)$-spherical Fourier transform, the $(\sigma,t)$-Poisson transform, the $(\sigma,t)$-Helgason Fourier transform, its inverse transform and Plancherel's Theorem. In the limit of large $t,$ $t \rightarrow +\infty,$ the resulting hyperbolic harmonic analysis tends to the standard Euclidean harmonic analysis on ${\mathbb R}^n,$ thus unifying hyperbolic and Euclidean harmonic analysis.

## Citation

Milton Ferreira. "Harmonic Analysis on the Einstein Gyrogroup." J. Geom. Symmetry Phys. 35 21 - 60, 2014. https://doi.org/10.7546/jgsp-35-2014-21-60

## Information

Published: 2014
First available in Project Euclid: 27 May 2017

zbMATH: 1327.83007
MathSciNet: MR3363622
Digital Object Identifier: 10.7546/jgsp-35-2014-21-60