Harmonic Analysis on the Möbius Gyrogroup

Ferreira, Milton

http://hdl.handle.net/10400.8/3803

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Authors

Ferreira, Milton

Abstract(s)

In this paper, we propose to develop harmonic analysis on the Poincaré ball B, a model of then-dimensional real hyperbolic space. The Poincaré ball B is the open ball of the Euclidean n-space $\bkR^n$ with radius t >0, centered at the origin of $\bkR^n$ and equipped with Möbius addition, thus forming a Möbius gyrogroup where Möbius addition in the ball plays the role of vector addition in $\bkR^n.$ For any t>0 and an arbitrary parameter $\sigma \in \bkR$ 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 tends to the standard Euclidean harmonic analysis on $\bkR^n,$ thus unifying hyperbolic and Euclidean harmonic analysis. As an application, we construct diffusive wavelets on B.

Keywords

Möbius gyrogroup Helgason-Fourier transform Spherical functions Hyperbolic convolution Eigenfunctions of the Laplace-Beltrami-operator Diffusive wavelets