Please use this identifier to cite or link to this item: http://hdl.handle.net/10773/14179
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

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