## Helgason-Marchaud inversion formulas for Radon transforms

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- by Boris Rubin PDF
- Proc. Amer. Math. Soc.
**130**(2002), 3017-3023 Request permission

## Abstract:

Let $X$ be either the hyperbolic space $\mathbb {H} ^{n}$ or the unit sphere $S^{n}$, and let $\Xi$ be the set of all $k$-dimensional totally geodesic submanifolds of $X, 1 \le k \le n-1$. For $x \in X$ and $\xi \in \Xi$, the totally geodesic Radon transform $f(x) \to \hat f(\xi )$ is studied. By averaging $\hat f(\xi )$ over all $\xi$ at a distance $\theta$ from $x$, and applying Riemann-Liouville fractional differentiation in $\theta$, S. Helgason has recovered $f(x)$. We show that in the hyperbolic case this method blows up if $f$ does not decrease sufficiently fast. The situation can be saved if one employs Marchaud’s fractional derivatives instead of the Riemann-Liouville ones. New inversion formulas for $\hat f(\xi ), f \in L^{p}(X)$, are obtained.## References

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## Additional Information

**Boris Rubin**- Affiliation: Institute of Mathematics, Hebrew University, Jerusalem 91904, Israel
- MR Author ID: 209987
- Email: boris@math.huji.ac.il
- Received by editor(s): May 16, 2001
- Published electronically: May 8, 2002
- Additional Notes: This work was partially supported by the Edmund Landau Center for Research in Mathematical Analysis and Related Areas, sponsored by the Minerva Foundation (Germany).
- Communicated by: David Preiss
- © Copyright 2002 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**130**(2002), 3017-3023 - MSC (2000): Primary 44A12; Secondary 52A22
- DOI: https://doi.org/10.1090/S0002-9939-02-06554-1
- MathSciNet review: 1908925