## 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