Abstract
We describe the range of the spherical Radon transform which evaluates integrals of a function in IRn 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.
Similar content being viewed by others
References
M. Agranovsky, D. Finch and P. Kuchment, Range condition for a spherical mean transform, Inverse Probl. and Imaging 3 (2009), 373–382.
M. Agranovsky, P. Kuchment and L. Kunyansky, On reconstruction formulas and algorithms for the thermoacoustic tomography in Photoacoustic Imaging and Spectroscopy (L. H. Wang, ed.) CRC Press, Boca Raton, FL, 2009, pp. 89–101.
M. Agranovsky, P. Kuchment and E. T. Quinto, Range descriptions for the spherical mean Radon transform, J. Funct. Anal. 248 (2007), 344–386.
M. L. Agranovsky and E. T. Quinto, Injectivity sets for the Radon transform over circles and complete systems of radial functions, J. Funct. Anal. 139 (1996), 383–414.
G. Ambartsoumian and P. Kuchment, A range description for the planar circular Radon transform, SIAM J. Math. Anal. 38 (2006), 681–692 (electronic).
G. Beylkin, The fundamental identity for iterated spherical means and the inversion formula for diffraction tomography and inverse scattering, J. Math. Phys. 24 (1983), 1399–1400.
G. Beylkin, Iterated spherical means in linearized inverse problems, in Conference on Inverse Scattering: Theory and Application (Tulsa, Okla., 1983), SIAM, Philadelphia, PA, 1983, pp. 112–117.
R. Courant and D. Hilbert, Methods of Mathematical Physics. Vol. II: Partial Differential Equations, Interscience, New York-London, 1962.
C. L. Epstein and B. Kleiner, Spherical means in annular regions, Comm. Pure Appl. Math. 46 (1993), 441–451.
D. Finch, M. Haltmeier and Rakesh, Inversion of spherical means and the wave equation in even dimensions, SIAM J. Appl. Math. 68 (2007), 392–412.
D. Finch, S. K. Patch and Rakesh, Determining a function from its mean values over a family of spheres, SIAM J. Math. Anal. 35 (2004), 1213–1240.
D. Finch and Rakesh, The range of the spherical mean value operator for functions supported in a ball, Inverse Problems 22 (2006), 923–938.
D. Finch and Rakesh, The spherical mean operator with centers on a sphere, Inverse Problems 23 (2007), 37–60.
I. M. Gelfand, S. G. Gindikin and M. I. Graev, Selected Topics in Integral Geometry, Amer. Math. Soc., Providence, RI, 2003.
F. John, Plane Waves and Spherical Means Applied to Partial Differential Equations, Springer-Verlag, New York, 1981. Reprint of the 1955 original.
P. K. Kuchment and L. Kunyansky, Mathematics of thermoacoustic tomography, Eur. J. Appl. Math. 19 (2008), 191–224.
V. Y. Lin and A. Pinkus, Approximation of multivariate functions, in Advances in Computational Mathematics (New Delhi, 1993), World Sci. Publ., River Edge, NJ, 1994, pp. 257–265.
S. K. Patch, Thermoacoustic tomography-consistency conditions and the partial scan problem, Phys. Med. Biol. 49 (2004), 1–11.
D. P. Zhelobenko, Compact Groups and their Representations, Amer. Math. Soc., Providence, RI, 1973.
Author information
Authors and Affiliations
Corresponding author
Additional information
partially supported by a grant from the ISF grant 688/08
partially supported by NSF DMS grants 0604778 and 0715090
Rights and permissions
About this article
Cite this article
Agranovsky, M., Nguyen, L.V. Range conditions for a spherical mean transform and global extendibility of solutions of the darboux equation. JAMA 112, 351–367 (2010). https://doi.org/10.1007/s11854-010-0033-0
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11854-010-0033-0