The support theorem for the single radius spherical mean transform
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© 2011, Razmadze Mathematical Institute. All rights reserved. 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 noninjective (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 [12], [13]. 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 Rconvex domain. We provide a simplified and selfcontained proof of this statement.
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Agranovsky, M., & Kuchment, P.
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35l05, 92c55, 65r32, 44a12

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