Untitled

Funder

Organizational Unit

Publications

Harmonic Analysis on the Möbius Gyrogroup

Publication . Ferreira, Milton

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.

2015-04Journal article

Open access

Harmonic Analysis on the Einstein Gyrogroup

Publication . Ferreira, Milton

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 $$ L_{\sigma,t} = \disp \left( 1 - \frac{\|x\|^2}{t^2} \right) \!\left( \Delta - \sum_{i,j=1}^n \frac{x_i x_j}{t^2} \frac{\partial^2}{\partial x_i \partial x_j} - \frac{\kappa}{t^2} \sum_{i=1}^n x_i \frac{\partial}{\partial x_i} + \frac{\kappa(2-\kappa)}{4t^2} \right)$$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 $(\sigma,t)$-convo\-lution, 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.

2014Journal article

Restricted access