Journal Title
Title of Journal: J Fourier Anal Appl
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Abbravation: Journal of Fourier Analysis and Applications
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Authors: Milton Ferreira
Publish Date: 2014/11/26
Volume: 21, Issue: 2, Pages: 281-317
Abstract
In this paper we propose to develop harmonic analysis on the Poincaré ball mathbb B tn a model of the ndimensional real hyperbolic space The Poincaré ball mathbb B tn is the open ball of the Euclidean nspace mathbb Rn with radius t 0 centered at the origin of mathbb Rn and equipped with Möbius addition thus forming a Möbius gyrogroup where Möbius addition in the ball plays the role of vector addition in mathbb Rn For any t0 and an arbitrary parameter sigma in mathbb R we study the sigma ttranslation the sigma tconvolution the eigenfunctions of the sigma tLaplace–Beltrami operator the sigma tHelgason 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 tn tends to the standard Euclidean harmonic analysis on mathbb Rn thus unifying hyperbolic and Euclidean harmonic analysis As an application we construct diffusive wavelets on mathbb B tnThis work was supported by Portuguese funds through the CIDMA Center for Research and Development in Mathematics and Applications and the Portuguese Foundation for Science and Technology “FCT Fundação para a Ciência e a Tecnologia” within project PEstOE/MAT/UI4106/2014
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