Invariant Radon transforms on a symmetric space
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- by Jeremy Orloff PDF
- Trans. Amer. Math. Soc. 318 (1990), 581-600 Request permission
Abstract:
Injectivity and support theorems are proved for a class of Radon transforms, ${R_\mu }$, for $\mu$ a smooth family of measures defined on a certain space of affine planes in ${\mathbb {X}_0}$, where ${\mathbb {X}_0}$ is the tangent space, of a Riemannian symmetric space of rank one. The transforms are defined by integrating against $\mu$ over these planes. We show that if ${R_\mu }f$ is supported inside a ball of radius $R$ then so is $f$. This is true for $f \in L_c^2({\mathbb {X}_0})$ or $f \in \mathcal {E}’({\mathbb {X}_0})$. Furthermore, ${R_\mu }$ is invertible on either of these domains. The main technique is to use facts about spherical harmonics to reduce the problem to a one-dimensional integral equation.References
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Additional Information
- © Copyright 1990 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 318 (1990), 581-600
- MSC: Primary 44A15; Secondary 43A85
- DOI: https://doi.org/10.1090/S0002-9947-1990-0958898-2
- MathSciNet review: 958898