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    Harmonic analysis on the Möbius gyrogroup

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    In this paper we propose to develop harmonic analysis on the Poincaré ball BtnB_t^n, a model of the n-dimensional real hyperbolic space. The Poincaré ball BtnB_t^n is the open ball of the Euclidean n-space RnR^n with radius t>0t>0, centered at the origin of RnR^n and equipped with Möbius addition, thus forming a Möbius gyrogroup where Möbius addition in the ball plays the role of vector addition in Rn\mathbb{R}^n. For any t>0t>0 and an arbitrary parameter σR\sigma \in R we study the (σ,t)(\sigma,t)-translation, the (σ,t)( \sigma,t)-convolution, the eigenfunctions of the (σ,t)(\sigma,t)-Laplace-Beltrami operator, the (σ,t)(\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+t \rightarrow +\infty the resulting hyperbolic harmonic analysis on BtnB_t^n tends to the standard Euclidean harmonic analysis on RnR^n, thus unifying hyperbolic and Euclidean harmonic analysis. As an application we construct diffusive wavelets on BtnB_t^n

    Ueber endlichgleiche Polyeder

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    Aristophanes, Knights

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