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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

The Radon transform on $ {\rm SL}(2,{\bf R})/{\rm SO}(2,{\bf R})$


Authors: D. I. Wallace and Ryuji Yamaguchi
Journal: Trans. Amer. Math. Soc. 297 (1986), 305-318
MSC: Primary 22E30; Secondary 44A15, 53C65
MathSciNet review: 849481
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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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DOI: http://dx.doi.org/10.1090/S0002-9947-1986-0849481-7
PII: S 0002-9947(1986)0849481-7
Article copyright: © Copyright 1986 American Mathematical Society