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Conic injectivity sets for the Radon transformation on spheres


Authors: V. V. Volchkov and Vit. V. Volchkov
Translated by: A. Plotkin
Original publication: Algebra i Analiz, tom 27 (2015), nomer 5.
Journal: St. Petersburg Math. J. 27 (2016), 709-730
MSC (2010): Primary 44A12
DOI: https://doi.org/10.1090/spmj/1413
Published electronically: July 26, 2016
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Abstract | References | Similar Articles | Additional Information

Abstract: The problem under study concerns description of nonzero functions that have zero integrals over all spheres with centers in a given set. For the corresponding integral transformation (Radon transformation on spheres), the kernel is described, and sharp uniqueness theorems are obtained. Applications of the main results to partial differential equations are considered: new uniqueness theorems are proved for the Darboux equation and the wave equation.


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  • 1. A. B. Aleksandrov, $ A$-integrability of boundary values of harmonic functions, Mat. Zametki 30 (1981), no. 1, 59-72; English transl., Math. Notes 30 (1981), no. 1, 515-523. MR 627941
  • 2. S. Helgason, Groups and geometric analysis. Integral geometry, invariant differential operators and special functions, Math. Surveys Monogr., vol. 83, Amer. Math. Soc., Providence, RI, 2000. MR 1790156
  • 3. V. V. Volchkov, Integral geometry and convolution equations, Kluwer. Acad. Publ., Dordrecht, 2003. MR 2016409
  • 4. F. John, Plane waves and spherical means applied to partial differential equations, Intersci. Publ., New York-London, 1955. MR 0075429
  • 5. R. Courant, D. Hilbert, Methods of mathematical physics, Vol. II. Partial differential equations, Intersci. Publ., New York-London, 1962. MR 0140802
  • 6. V. V. Volchkov, Injectivity sets for the Radon transform on spheres, Izv. Ross. Akad. Nauk Ser. Mat. 63 (1999), no. 3, 63-76; English transl., Izv. Math. 63 (1999), no. 3, 481-493. MR 1712132
  • 7. M. L. Agranovsky, C. A. Berenstein, and P. Kuchment, Approximation by spherical waves in $ L^p$ spaces, J. Geom. Anal. 6 (1996), no. 3, 365-383. MR 1471897
  • 8. 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), no. 2, 383-414. MR 1402770
  • 9. M L. Agranovsky, V. V. Volchkov, and L. A. Zalcman, Conical injectivity sets for the spherical Radon transform, Bull. London Math. Soc. 31 (1999), no. 2, 231-236. MR 1664137
  • 10. D. H. Armitage, Cones on which entire harmonic functions can vanish, Proc. Roy. Irish Acad. Sect. A 92 (1992), no. 1, 107-110. MR 1173388 (93h:31003)
  • 11. V. P. Burskiĭ, Investigation methods of boundary value problems for general differential equations, Kiev, Naukova Dumka, 2002. (Russian)
  • 12. Vit. V. Volchkov, A local two-radius theorem on the sphere, Algebra i Analiz 16 (2004), no. 3, 24-55; English transl., St. Petersburg Math. J. 16 (2005), no. 3, 453-475. MR 2083565
  • 13. E. M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Math. Ser., vol. 32, Princeton Univ. Press, Princeton, NS, 1971. MR 0304972
  • 14. A. Erdelyi, F. Oberheftinger, and F. Triconi, Higher transcendental functions. Vol. I, II, Based in part on notes left by H. Bateman, McGraw-Hell Book Co., New-York, 1953. MR 0058756
  • 15. S. Helgason, Integral geometry and radon transforms, Springer, New York, 2011. MR 2743116
  • 16. V. V. Volchkov and Vit. V. Volchkov, Harmonic analysis of mean periodic functions on symmetric spaces and the Heisenberg group, Springer Monogr. Math., Springer-Verlag, London, 2009. MR 2527108 (2011f:43021)
  • 17. S. Lang, $ SL_2(R)$, Addison-Wesley Publ. Co., Reading, Mass, 1975. MR 0430163
  • 18. N. Ya. Vilenkin, Special functions and the theory of group representations, Nauka, Moscow, 1991; English transl. first. edition, Transl. Math. Monogr., vol. 22, Amer. Math. Soc., Providence, RI, 1968. MR 0229863 (37:5429), MR 1177172 (93d:33013)
  • 19. A. N. Kolmogorov and S. V. Fomin, Elements of the theory of functions and functional analysis, Nauka, Moscow, 1989; English transl. first edition, Graylock Press, Rochester, N.Y., 1957. MR 0085462 (19:44d); MR 1025126 (90k:46001)
  • 20. L. Hörmander, The analysis of linear partial differential operators. I. Distribution theory and Fourier analysis, 2nd ed., Grundlehren Math. Wiss., Bd. 256, Springer-Verlag, Berlin, 1990. MR 1065993 (91m:35001a)
  • 21. Ya. B. Lopatinskiĭ, Introduction to the modern theory of partial differential equations, Kiev, Nakova Dumka, 1980. (Russian). MR 591676

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Additional Information

V. V. Volchkov
Affiliation: Donetsk national university, 24 Universitetskaya str., Donetsk 83001, Ukraine
Email: valeriyvolchkov@gmail.com

Vit. V. Volchkov
Affiliation: Donetsk national university, 24 Universitetskaya str., Donetsk 83001, Ukraine

DOI: https://doi.org/10.1090/spmj/1413
Keywords: Conic sets, Radon transformation, Darboux equation, wave equation
Received by editor(s): December 16, 2014
Published electronically: July 26, 2016
Article copyright: © Copyright 2016 American Mathematical Society

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