Mathematical Notes

, Volume 104, Issue 5–6, pp 689–695 | Cite as

On the Completeness of Products of Harmonic Functions and the Uniqueness of the Solution of the Inverse Acoustic Sounding Problem

  • M. Yu. Kokurin


It is proved that the family of all pairwise products of regular harmonic functions on D and of the Newtonian potentials of points on the line L ⊂ Rn is complete in L2(D), where D is a bounded domain in Rn, n ≥ 3, such that \(\bar D\)L = ∅. This result is used in the proof of uniqueness theorems for the inverse acoustic sounding problem in R3.


harmonic function Newtonian potential completeness integral equation acoustic sounding inverse problem unique solvability 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    A. G. Ramm, Multidimensional Inverse Scattering Problems (John Wiley, New York, 1992;Mir, Moscow, 1994).zbMATHGoogle Scholar
  2. 2.
    V. Isakov, Inverse Problems for Partial Differential Equations (Springer, New York, 2006).zbMATHGoogle Scholar
  3. 3.
    C. Kenig and M. Salo, “Recent progress in the Calderón problem with partial data,” in Inverse Problems and Applications, Contemp. Math. (Amer.Math. Soc., Providence, RI, 2014), Vol. 615, pp. 193–222.zbMATHGoogle Scholar
  4. 4.
    E. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces (Princeton Univ. Press, Princeton, NJ, 1971; Mir, Moscow, 1974).zbMATHGoogle Scholar
  5. 5.
    V. A. Ditkin and A. P. Prudnikov, Integral Transformations and Operational Calculus (Fizmatgiz, Moscow, 1961) [in Russian].Google Scholar
  6. 6.
    I. A. Kipriyanov, Singular Elliptic Boundary-Value Problems (Nauka, Moscow, 1997) [in Russian].zbMATHGoogle Scholar
  7. 7.
    L. Schwartz, Analyse Mathématique (Hermann, Paris, 1967;Mir, Moscow, 1972).zbMATHGoogle Scholar
  8. 8.
    N. I. Akhiezer, Lectures in Approximation Theory (Nauka, Moscow, 1965) [in Russian].Google Scholar
  9. 9.
    M. M. Lavrent’ev, “On an inverse problem for the wave equation, Dokl. Akad. Nauk SSSR 157 (3), 520–521 (1964).Google Scholar
  10. 10.
    A. B. Bakushinskii, A. I. Kozlov, and M. Yu. Kokurin, “On some inverse problem for a three-dimensional wave equation,” Zh. Vychisl.Mat. Mat. Fiz. 43 (8), 1201–1209 (2003) [Comput.Math. Math. Phys. 43 (8), 1149–1158 (2003)].MathSciNetzbMATHGoogle Scholar
  11. 11.
    M. M. Lavrent’ev, “A class of inverse problems for differential equations,” Dokl. Akad. Nauk SSSR 160 (1), 32–35 (1965).MathSciNetzbMATHGoogle Scholar
  12. 12.
    M. M. Lavrent’ev, V.G. Romanov, and S. P. Shishatskii, ill-Posed Problems ofMathematical Physics and Analysis (Nauka, Moscow, 1980) [in Russian].Google Scholar
  13. 13.
    M. Yu. Kokurin, “On a multidimensional integral equation with data supported by low-dimensional analytic manifolds,” J. Inverse Ill-Posed Probl. 21 (1), 125–140 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    M. Yu. Kokurin, “Sets of uniqueness for harmonic and analytic functions and inverse problems for wave equations,” Mat. Zametki 97 (3), 397–406 (2015) [Math. Notes 97 (3), 376–383 (2015)].MathSciNetCrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Mari State UniversityIoshkar-OlaRussia

Personalised recommendations