Abstract
We describe the range of the spherical Radon transform which evaluates integrals of a function in ℝn over all spheres centered on a given sphere. Such a transform attracts much attention due to its applications in approximation theory and (thermo- and photoacoustic) tomography. Range descriptions for this transform have been obtained recently. They include two types of conditions: an orthogonality condition and, for even n, a moment condition. It was later discovered that, in all dimensions, the moment condition follows from the orthogonality condition (and can therefore can be dropped). In terms of the Darboux equation, which describes spherical means, this indirectly implies that solutions of certain boundary value problems in a domain extend automatically outside of the domain. In this article, we present a direct proof of this global extendibility phenomenon for the Darboux equation. Correspondingly, we deliver an alternative proof of the range characterization theorem.
| Original language | English |
|---|---|
| Pages (from-to) | 351-367 |
| Number of pages | 17 |
| Journal | Journal d'Analyse Mathematique |
| Volume | 112 |
| Issue number | 1 |
| DOIs | |
| State | Published - 2010 |
Bibliographical note
Funding Information:∗partially supported by a grant from the ISF grant 688/08 †partially supported by NSF DMS grants 0604778 and 0715090
Funding Information:
This completes the proof of Lemma 4.2 and thus finishes the proof of Theorem 2.1. □ Acknowledgments. This work was done while the first author was on sabbatical leave at Department of Mathematics, Texas A&M University. He thanks the ISF for support and the department for its hospitality and excellent working conditions. The second author thanks the NSF for its support. Both authors thank Peter Kuchment for useful discussions and valuable remarks.
Funding
∗partially supported by a grant from the ISF grant 688/08 †partially supported by NSF DMS grants 0604778 and 0715090 This completes the proof of Lemma 4.2 and thus finishes the proof of Theorem 2.1. □ Acknowledgments. This work was done while the first author was on sabbatical leave at Department of Mathematics, Texas A&M University. He thanks the ISF for support and the department for its hospitality and excellent working conditions. The second author thanks the NSF for its support. Both authors thank Peter Kuchment for useful discussions and valuable remarks.
| Funders | Funder number |
|---|---|
| NSF DMS | 0604778, 0715090 |
| National Science Foundation | |
| Israel Science Foundation |