The Radon transform on 𝑆𝐿(2,𝑅)/𝑆𝑂(2,𝑅)

Wallace, D. I.; Yamaguchi, Ryuji

doi:10.1090/S0002-9947-1986-0849481-7

The Radon transform on $\textrm {SL}(2,\textbf {R})/\textrm {SO}(2,\textbf {R})$
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by D. I. Wallace and Ryuji Yamaguchi PDF

Trans. Amer. Math. Soc. 297 (1986), 305-318 Request permission

Abstract:

Let $G$ be $SL(2,{\mathbf {R}})$. $G$ acts on the upper half-plane $\mathcal {H}$ by the Möbius transformation, providing $\mathcal {H}$ with the Riemannian metric structure along with the Laplacian, $\Delta$. We study the integral transform along each geodesic. $G$ acts on $\mathcal {P}$, the space of all geodesics, in a natural way, providing $\mathcal {P}$ with its invariant measure and its own Laplacian. ($\mathcal {P}$ actually is a coset space of $G$.) Therefore the above transform can be viewed as a map from a suitable function space on $\mathcal {H}$ to a suitable function space on $\mathcal {P}$. We prove a number of properties of this transform, including the intertwining properties with its Laplacians and its relation to the Fourier transforms.

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