Ferreira, Milton2019-02-062019-02-062015-04Ferreira M., Harmonic analysis on the Möbius gyrogroup, J. Fourier Anal. Appl., 21(2), 2015, 281-3171069-58691531-5851http://hdl.handle.net/10400.8/3803In 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.engMöbius gyrogroupHelgason-Fourier transformSpherical functionsHyperbolic convolutionEigenfunctions of the Laplace-Beltrami-operatorDiffusive waveletsjournal article10.1007/s00041-014-9370-1