Helgason-Marchaud inversion formulas for Radon transforms

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...

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Bibliographic Details
Published in:Proceedings of the American Mathematical Society 2002-10, Vol.130 (10), p.3017-3023
Main Author: Rubin, Boris
Format: Article
Language:English
Subjects:
Applied mathematics
Convex and discrete geometry
Exact sciences and technology
Geometry
Integral transforms, operational calculus
Mathematical analysis
Mathematical integrals
Mathematical theorems
Mathematics
Radon
Sciences and techniques of general use
Wavelet analysis
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Summary: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.
ISSN:0002-9939
1088-6826
DOI:10.1090/S0002-9939-02-06554-1