Skip to main content
SpringerLink
Log in

Range description for a spherical mean transform on spaces of constant curvature

  • Published:
Journal d'Analyse Mathématique Aims and scope

Abstract

Let X be a Riemannian manifold and R be the spherical mean transform in X. Let S be a geodesic sphere in X and R S be the restriction of R to the set of geodesic spheres centered on S. We present a complete range description for R S when X is either the hyperbolic space H n or the sphere S n (n ≥ 2 in both cases). The description is analogous to a result for the euclidean space ℝn obtained by M. Agranovsky, D. Finch, and P. Kuchment and by M. Agranovsky and L. V. Nguyen.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Mark Agranovsky, Carlos Berenstein, and Peter Kuchment, Approximation by spherical waves in Lp-spaces, J. Geom. Anal. 6 (1996), 365–383.

    Article  MathSciNet  MATH  Google Scholar 

  2. Mark Agranovsky, Peter Kuchment, and Leonid Kunyansky, On reconstruction formulas and algorithms for the thermoacoustic tomography, Photoacoustic Imaging and Spectroscopy, CRC Press, Boca Raton, FL, 2009, pp. 89–101.

    Chapter  Google Scholar 

  3. Mark Agranovsky, Peter Kuchment, and Eric Todd Quinto, Range descriptions for the spherical mean Radon transform, J. Funct. Anal. 248 (2007), 344–386.

    Article  MathSciNet  MATH  Google Scholar 

  4. Mark Agranovsky and Linh V. Nguyen, Range conditions for a spherical mean transform and global extendibility of solutions of the Darboux equation, J. Anal. Math. 112 (2010), 351–367.

    Article  MathSciNet  MATH  Google Scholar 

  5. Mark L. Agranovsky and Eric Todd Quinto, Injectivity sets for the Radon transform over circles and complete systems of radial functions, J. Funct. Anal. 139 (1996), 383–414.

    Article  MathSciNet  MATH  Google Scholar 

  6. Yuri A. Antipov, Ricardo Estrada, and Boris Rubin, Method of analytic continuation for the inverse spherical mean transform in constant curvature spaces, J. Anal. Math. 118 (2012).

  7. Mark Agranovsky, David Finch, and Peter Kuchment, Range condition for a spherical mean transform, Inverse Probl. Imaging 3 (2009), 373–382.

    Article  MathSciNet  MATH  Google Scholar 

  8. Gaik Ambartsoumian and Peter Kuchment, A range description for the planar circular Radon transform, SIAM J. Math. Anal. 38 (2006), 681–692.

    Article  MathSciNet  MATH  Google Scholar 

  9. 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.

    Article  MathSciNet  Google Scholar 

  10. Gregory Beylkin, Iterated spherical means in linearized inverse problems, Conference on Inverse Scattering: Theory and Application, SIAM, Philadelphia, PA, 1983, pp. 112–117.

    Google Scholar 

  11. A. M. Cormack and E. T. Quinto, A Radon transform on spheres through the origin in R n and applications to the Darboux equation, Trans. Amer.Math. Soc. 260 (1980), 575–581.

    MathSciNet  MATH  Google Scholar 

  12. Charles L. Epstein and Bruce Kleiner, Spherical means in annular regions, Comm. Pure Appl. Math. 46 (1993), 441–451.

    Article  MathSciNet  MATH  Google Scholar 

  13. Lawrence C. Evans, Partial Differential Equations, American Mathematical Society, Providence, RI, 1998.

    MATH  Google Scholar 

  14. David Finch, Markus Haltmeier, and Rakesh, Inversion of spherical means and the wave equation in even dimensions, SIAM J. Appl. Math. 68 (2007), 392–412.

    Article  MathSciNet  MATH  Google Scholar 

  15. David Finch, Sarah K. Patch, and Rakesh, Determining a function from its mean values over a family of spheres, SIAM J. Math. Anal. 35 (2004), 1213–1240.

    Article  MathSciNet  MATH  Google Scholar 

  16. David Finch and Rakesh, Trace identities for solutions of the wave equation with initial data supported in a ball, Math. Methods Appl. Sci. 28 (2005), 1897–1917.

    Article  MathSciNet  MATH  Google Scholar 

  17. David Finch and Rakesh, The range of the spherical mean value operator for functions supported in a ball, Inverse Problems 22 (2006), 923–938.

    Article  MathSciNet  MATH  Google Scholar 

  18. David Finch and Rakesh, The spherical mean value operator with centers on a sphere, Inverse Problems 23 (2007), S37–S49.

    Article  MathSciNet  MATH  Google Scholar 

  19. I.M. Gelfand, S. G. Gindikin, andM. I. Graev, Selected Topics in Integral Geometry, American Mathematical Society, Providence, RI, 2003.

    MATH  Google Scholar 

  20. M. Haltmeier, Universal inversion formulas for recovering a function from spherical means, SIAM J. Math. Anal. 46 (2014), 214–232.

    Article  MathSciNet  MATH  Google Scholar 

  21. Sigurdur Helgason, The Radon Transform, Birkhäuser Boston, MA., 1980.

    Book  MATH  Google Scholar 

  22. Sigurdur Helgason, Groups and Geometric Analysis, Academic Press Inc., Orlando, FL, 1984.

    MATH  Google Scholar 

  23. Fritz John, Plane Waves and Spherical Means Applied to Partial Differential Equations, Springer-Verlag, New York, 1981.

  24. Peter Kuchment and Leonid Kunyansky, Mathematics of thermoacoustic tomography, European J. Appl. Math. 19 (2008), 191–224.

    Article  MathSciNet  MATH  Google Scholar 

  25. Leonid A. Kunyansky, Explicit inversion formulae for the spherical mean Radon transform, Inverse Problems 23 (2007), 373–383.

    Article  MathSciNet  MATH  Google Scholar 

  26. Leonid Kunyansky, Reconstruction of a function from its spherical (circular) means with the centers lying on the surface of certain polygons and polyhedra, Inverse Problems 27 (2011), 025012.

    Article  MathSciNet  MATH  Google Scholar 

  27. Vladimir Ya. Lin and Allan Pinkus, Fundamentality of ridge functions, J. Approx. Theory 75 (1993), 295–311.

    Article  MathSciNet  MATH  Google Scholar 

  28. Vladimir Ya. Lin and Allan Pinkus, Approximation of multivariate functions, Advances in Computational Mathematics, World Sci. Publ., River Edge, NJ, 1994, pp. 257–265.

    Google Scholar 

  29. F. Natterer, Photo-acoustic inversion in convex domains, Inverse Probl. Imaging 6 (2012), 1–6.

    Article  MathSciNet  MATH  Google Scholar 

  30. Linh V. Nguyen, A family of inversion formulas in thermoacoustic tomography, Inverse Probl. Imaging 3 (2009), 649–675.

    Article  MathSciNet  MATH  Google Scholar 

  31. Linh V. Nguyen, Spherical mean transform: a pde approach, Inverse Probl. Imaging 7 (2013), 243–252.

    Article  MathSciNet  MATH  Google Scholar 

  32. Gestur Ólafsson and Henrik Schlichtkrull, A local Paley-Wiener theorem for compact symmetric spaces, Adv. Math. 218 (2008), 202–215.

    Article  MathSciNet  MATH  Google Scholar 

  33. Victor Palamodov, Remarks on the general Funk transform and thermoacoustic tomography, Inverse Probl. Imaging 4 (2010).

  34. V. P. Palamodov, A uniform reconstruction formula in integral geometry, Inverse Problems 28 (2012), 065014.

    Article  MathSciNet  MATH  Google Scholar 

  35. Robert S. Strichartz, Local harmonic analysis on spheres, J. Funct. Anal. 77 (1988), 403–433.

    Article  MathSciNet  MATH  Google Scholar 

  36. Y. Salman, An inversion formula for the spherical mean transform with data on an ellipsoid in two and three dimensions, J. Math. Anal. Appl. 420 (2014), 612–620.

    Article  MathSciNet  MATH  Google Scholar 

  37. V. V. Volchkov, Spherical means on symmetric spaces, Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki 2002, no. 3, 15–19.

  38. V. V. Volchkov, Integral Geometry and Convolution Equations, Kluwer Academic Publishers, Dordrecht, 2003.

    Book  MATH  Google Scholar 

  39. Lawrence Zalcman, Offbeat integral geometry, Amer. Math. Monthly 87 (1980), 161–175.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Idaho, USA, Moscow, Idaho, 83843

    Linh V. Nguyen

Authors
  1. Linh V. Nguyen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Linh V. Nguyen.

Additional information

The research is partially supported by NSF DMS grant # 1212125.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Nguyen, L.V. Range description for a spherical mean transform on spaces of constant curvature. JAMA 128, 191–214 (2016). https://doi.org/10.1007/s11854-016-0006-z

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11854-016-0006-z

Keywords

Search

Navigation