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Title: | Harmonic analysis on the Möbius gyrogroup |
Author: | Ferreira, Milton |
Keywords: | Möbius gyrogroup Helgason-Fourier transform Spherical functions Hyperbolic convolution Eigenfunctions of the Laplace-Beltrami-operator Diffusive wavelets |
Issue Date: | Apr-2015 |
Publisher: | Springer |
Abstract: | In this paper we propose to develop harmonic analysis on the Poincaré ball $B_t^n$, a model of the n-dimensional real hyperbolic space. The Poincaré ball $B_t^n$ is the open ball of the Euclidean n-space $R^n$ with radius $t>0$, centered at the origin of $R^n$ and equipped with Möbius addition, thus forming a Möbius gyrogroup where Möbius addition in the ball plays the role of vector addition in $\mathbb{R}^n$. For any $t>0$ and an arbitrary parameter $\sigma \in R$ we study the $(\sigma,t)$-translation, the $( \sigma,t)$-convolution, the eigenfunctions of the $(\sigma,t)$-Laplace-Beltrami operator, the $(\sigma,t)$-Helgason Fourier transform, its inverse transform and the associated Plancherel's Theorem, which represent counterparts of standard tools, thus, enabling an effective theory of hyperbolic harmonic analysis. Moreover, when $t \rightarrow +\infty$ the resulting hyperbolic harmonic analysis on $B_t^n$ tends to the standard Euclidean harmonic analysis on $R^n$, thus unifying hyperbolic and Euclidean harmonic analysis. As an application we construct diffusive wavelets on $B_t^n$. |
Peer review: | yes |
URI: | http://hdl.handle.net/10773/14179 |
DOI: | 10.1007/s00041-014-9370-1 |
ISSN: | 1531-5851 |
Appears in Collections: | CIDMA - Artigos CHAG - Artigos |
Files in This Item:
File | Description | Size | Format | |
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HAMG_MFerreira_2015.pdf | Accepted author's manuscript | 490.66 kB | Adobe PDF | View/Open |
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