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Approximation by spherical waves inL p-spaces

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Abstract

The paper is devoted to the following problem. Consider the set of all radial functions with centers at the points of a closed surface inR n. Are such functions complete in the spaceL q(R n)? It is shown that the answer is positive if and only ifq is not less than 2n/(n + 1). A similar question is also answered for Riemannian symmetric spaces of rank 1. Relations of this problem with the wave and heat equations are also discussed.

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References

  1. Agranovsky, M. L., and Quinto, E. T. Injectivity sets for the Radon transform over circles and complete systems of radial functions.J. of Functional Anal. 139(2), 383–414 (1966).

    Article  MathSciNet  Google Scholar 

  2. Agranovsky, M. L., and Quinto, E. T. Injectivity sets for the Radon transform over circles and complete systems of radial functions: an announcement.Intern. Math. Research Notes 11 467–473 (1994).

    Article  MathSciNet  Google Scholar 

  3. Courant, R., and Hilbert, D.Methods of Mathematical Physics, vol. 2. Interscience, New York, 1953.

    Google Scholar 

  4. Edwards, D. E., and Evans, W. D.Spectral Theory and Differential Operators. Oxford Science Publ., 1990.

  5. Ehrenpreis, L.The Radon Transform, preprint of book, 229 pp.

  6. Helgason, S.Geometric Analysis on Symmetric Spaces. Amer. Math. Soc., Providence, 1994.

    MATH  Google Scholar 

  7. Lin, V. Ya., and Pinkus, A. Fundamentality of ridge functions.J. Approx. Theory 75, 295–311 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  8. Lin, V. Ya., and Pinkus, A. Approximation of multivariable functions. InAdvances in Computational Mathematics (H. P. Dikshit and C. A. Micchelli, eds.). World Sci. Publ., 1994, 1–9.

  9. Rawat, R., and Sitaram, A. The injectivity of the Pompeiu transform andL p analogues of the Wiener Tauberian Theorem.Israel J. Math. 91, 307–316 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  10. Strichartz, R. Harmonic analysis as spectral theory of Laplacians.J. Fcnal. Anal. 87, 51–148 (1989).

    MathSciNet  MATH  Google Scholar 

  11. Volchkov, V. On some questions connected with the Pompeiu problem. Preprint, 1995.

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Authors and Affiliations

  1. Department of Mathematics, Bar-Ilan University, USA

    Mark Agranovsky, Carlos Berenstein & Peter Kuchment

  2. Department of Mathematics, University of Maryland, Institute for Systems Research, USA

    Mark Agranovsky, Carlos Berenstein & Peter Kuchment

  3. Department of Mathematics, Wichita State University, Wichita, Kansas

    Mark Agranovsky, Carlos Berenstein & Peter Kuchment

Authors
  1. Mark Agranovsky
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  2. Carlos Berenstein
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  3. Peter Kuchment
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Additional information

M.A. was partially supported by the Binational US-Israel Science Foundation, grant MCS 9123862. C.B. was partially supported by NSF grants DMS 9225043 and EEC 9402384 and the previous BNSF grant. P.K. was partially supported by an NSF EPSCoR Grant.

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Agranovsky, M., Berenstein, C. & Kuchment, P. Approximation by spherical waves inL p-spaces. J Geom Anal 6, 365–383 (1996). https://doi.org/10.1007/BF02921656

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  • DOI: https://doi.org/10.1007/BF02921656

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