Let f ∈ Lp(Rn) and R > 0. The transform is considered that integrates the function f over (almost) all spheres of radius R in Rn. This operator is known to be non-injective (as one can see by taking Fourier transform). However, the counterexamples that can be easily constructed using Bessel functions of the 1st kind, only belong to Lp if p > 2n/(n − 1). It has been shown previously by S. Thangavelu that for p not exceeding the critical number 2n/(n − 1), the transform is indeed injective. A support theorem that strengthens this injectivity result can be deduced from the results of , . Namely, if K is a convex bounded domain in Rn, the index p is not above 2n/(n − 1), and (almost) all the integrals of f over spheres of radius R not intersecting K are equal to zero, then f is supported in the closure of the domain K. In fact, convexity in this case is too strong a condition, and the result holds for any what we call R-convex domain. We provide a simplified and self-contained proof of this statement.
|Number of pages||16|
|Journal||Memoirs on Differential Equations and Mathematical Physics|
|State||Published - 2011|
Bibliographical noteFunding Information:
The work of the first author was performed when he was visiting Texas A&M University and was partially supported by the ISF (Israel Science Foundation) grant 688/08. The second author was partially supported by the NSF grants DMS 0604778 and 0908208 and by the IAMCS. The authors express their gratitude to ISF, NSF, Texas A&M University, and IAMCS for the support. We are also grateful to the reviewer of the first version of the text for pointing to us the reference to  and to Y. Lyubarskii for useful comments and references.
© 2011, Razmadze Mathematical Institute. All rights reserved.
- Radon transform
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