Journal d'Analyse Mathématique

, Volume 112, Issue 1, pp 351–367 | Cite as

Range conditions for a spherical mean transform and global extendibility of solutions of the darboux equation

Article

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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    M. Agranovsky, D. Finch and P. Kuchment, Range condition for a spherical mean transform, Inverse Probl. and Imaging 3 (2009), 373–382.MATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    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.CrossRefGoogle Scholar
  3. [3]
    M. Agranovsky, P. Kuchment and E. T. Quinto, Range descriptions for the spherical mean Radon transform, J. Funct. Anal. 248 (2007), 344–386.MATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    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.MATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    G. Ambartsoumian and P. Kuchment, A range description for the planar circular Radon transform, SIAM J. Math. Anal. 38 (2006), 681–692 (electronic).CrossRefMathSciNetGoogle Scholar
  6. [6]
    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.CrossRefMathSciNetGoogle Scholar
  7. [7]
    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.Google Scholar
  8. [8]
    R. Courant and D. Hilbert, Methods of Mathematical Physics. Vol. II: Partial Differential Equations, Interscience, New York-London, 1962.MATHGoogle Scholar
  9. [9]
    C. L. Epstein and B. Kleiner, Spherical means in annular regions, Comm. Pure Appl. Math. 46 (1993), 441–451.MATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    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.MATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    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.MATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    D. Finch and Rakesh, The range of the spherical mean value operator for functions supported in a ball, Inverse Problems 22 (2006), 923–938.MATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    D. Finch and Rakesh, The spherical mean operator with centers on a sphere, Inverse Problems 23 (2007), 37–60.CrossRefMathSciNetGoogle Scholar
  14. [14]
    I. M. Gelfand, S. G. Gindikin and M. I. Graev, Selected Topics in Integral Geometry, Amer. Math. Soc., Providence, RI, 2003.MATHGoogle Scholar
  15. [15]
    F. John, Plane Waves and Spherical Means Applied to Partial Differential Equations, Springer-Verlag, New York, 1981. Reprint of the 1955 original.MATHGoogle Scholar
  16. [16]
    P. K. Kuchment and L. Kunyansky, Mathematics of thermoacoustic tomography, Eur. J. Appl. Math. 19 (2008), 191–224.MATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    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.Google Scholar
  18. [18]
    S. K. Patch, Thermoacoustic tomography-consistency conditions and the partial scan problem, Phys. Med. Biol. 49 (2004), 1–11.CrossRefGoogle Scholar
  19. [19]
    D. P. Zhelobenko, Compact Groups and their Representations, Amer. Math. Soc., Providence, RI, 1973.MATHGoogle Scholar

Copyright information

© Hebrew University Magnes Press 2010

Authors and Affiliations

  1. 1.Department of MathematicsBar-Ilan UniversityRamat-GanIsrael
  2. 2.Department of MathematicsTexas A&M UniversityCollege StationUSA
  3. 3.Department of MathematicsUniversity of IdahoMoscowUSA

Personalised recommendations