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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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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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Additional Information
  • © Copyright 1986 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 297 (1986), 305-318
  • MSC: Primary 22E30; Secondary 44A15, 53C65
  • DOI: https://doi.org/10.1090/S0002-9947-1986-0849481-7
  • MathSciNet review: 849481