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Harmonic analysis on the Einstein gyrogroup. (English) Zbl 1327.83007

Summary: 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 generalized Laplace-Beltrami operator \[ L_{\sigma,t}= \Biggl(1-{\| x\|^2\over t^2}\Biggr)\Biggl(\Delta- \sum^n_{i,j=1} {x_ix_j\over t^2}{\partial^2\over\partial x_i\partial x_j}-{\kappa\over t^2} \sum^n_{i=1} x_i{\partial\over\partial x_i}+ {\kappa(2-\kappa)\over 4t^2}\Biggr), \] where \(\kappa= n+\sigma\) and \(\sigma\in\mathbb{R}\) is an arbitrary parameter. The generalized harmonic analysis for \(L_{\sigma,t}\) gives rise to the \((\sigma,t)\)-translation, the \((\sigma, t)\)-convolution, 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 \(t\), \(t\to+\infty\), the resulting hyperbolic harmonic analysis tends to the standard Euclidean harmonic analysis on \(\mathbb{R}^n\), thus unifying hyperbolic and Euclidean harmonic analysis.

MSC:

83A05 Special relativity
42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
44A35 Convolution as an integral transform
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