Harmonic Analysis on the Möbius Gyrogroup

Abstract

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