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http://hdl.handle.net/10773/14178
Title: | Harmonic analysis on the Einstein gyrogroup |
Author: | Ferreira, Milton |
Keywords: | Einstein gyrogroup Generalised Helgason-Fourier transform Spherical functions Hyperbolic convolution Eigenfunctions Laplace-Beltrami-operator |
Issue Date: | 2014 |
Publisher: | JGSP |
Abstract: | In this paper we study harmonic analysis on the Einstein gyrogroup of the open ball of R$^n$, $n \in N,$ centered at the origin and with arbitrary radius $t \in R^+,$ associated to the generalised Laplace-Beltrami operator $$ L_{\sigma,t} = \disp \left( 1 - \frac{\|x\|^2}{t^2} \right) \!\left( \Delta - \sum_{i,j=1}^n \frac{x_i x_j}{t^2} \frac{\partial^2}{\partial x_i \partial x_j} - \frac{\kappa}{t^2} \sum_{i=1}^n x_i \frac{\partial}{\partial x_i} + \frac{\kappa(2-\kappa)}{4t^2} \right)$$ 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 $R^n,$ thus unifying hyperbolic and Euclidean harmonic analysis. |
Peer review: | yes |
URI: | http://hdl.handle.net/10773/14178 |
DOI: | 10.7546/jgsp-35-2014-21-60 |
ISSN: | 1312-5192 |
Appears in Collections: | CIDMA - Artigos |
Files in This Item:
File | Description | Size | Format | |
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Harmonic_analysis_Einstein_gyrogroup_MFerreira.pdf | Accepted Author's manuscript | 484.52 kB | Adobe PDF | View/Open |
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