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Inversion Theorems for the local Pompeiu transformation in the quaternion hyperbolic space


Authors: Vit. V. Volchkov and N. P. Volchkova
Translated by: N. Yu. Netsvetaev
Original publication: Algebra i Analiz, tom 15 (2003), nomer 5.
Journal: St. Petersburg Math. J. 15 (2004), 753-771
MSC (2000): Primary 44A15, 53C65
DOI: https://doi.org/10.1090/S1061-0022-04-00830-1
Published electronically: July 29, 2004
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Abstract | References | Similar Articles | Additional Information

Abstract: A construction for inversion of the local Pompeiu transformation is obtained for the family consisting of two geodesic balls on the quaternion hyperbolic space.


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  • 1. C. A. Berenstein, R. Gay, and A. Yger, Inversion of the local Pompeiu transform, J. Anal. Math. 54 (1990), 259-287. MR 1041185 (91g:44004)
  • 2. C. A. Berenstein and D. Struppa, Complex analysis and convolution equations, Several Complex Variables. V, Itogi Nauki i Tekhniki Sovrem. Probl. Mat. Fund. Naprav., t. 54, VINITI, Moscow, 1989, pp. 5-111; English transl., Encyclopaedia Math. Sci., vol. 54, Springer-Verlag, Berlin, 1993, pp. 1-108. MR 1039621 (91d:32001)
  • 3. L. Zalcman, A bibliographic survey of the Pompeiu problem, Approximation by Solutions of Partial Differential Equations (Hanstholm, 1991), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 365, Kluwer Acad. Publ., Dordrecht, 1992, pp. 185-194. MR 1168719 (93e:26001)
  • 4. I. Netuka and J. Veselý, Mean value property and harmonic functions, Classical and Modern Potential Theory and Applications (Chateau de Bonas, 1993), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 430, Kluwer Acad. Publ., Dordrecht, 1994, pp. 359-398. MR 1321628 (96c:31001)
  • 5. L. Zalcman, Supplementary bibliography to : ``A bibliographic survey of the Pompeiu problem'', Radon Transforms and Tomography (South Hadley, MA, 2000), Contemp. Math., vol. 278, Amer. Math. Soc., Providence, RI, 2001, pp. 69-74. MR 1851479
  • 6. -, Analyticity and the Pompeiu problem, Arch. Rational Mech. Anal. 47 (1972), 237-254. MR 0348084 (50:582)
  • 7. C. A. Berenstein and A. Yger, Le problème de la déconvolution, J. Funct. Anal. 54 (1983), no. 2, 113-160. MR 0724701 (85i:46048)
  • 8. C. A. Berenstein and R. Gay, A local version of the two-circles theorem, Israel J. Math. 55 (1986), 267-288. MR 0876395 (88f:42044)
  • 9. V. V. Volchkov, Definitive version of a local two-radius theorem, Mat. Sb. 186 (1995), no. 6, 15-34; English transl., Sb. Math. 186 (1995), no. 6, 783-802. MR 1349012 (96i:43005)
  • 10. S. Helgason, Groups and geometric analysis. Integral geometry, invariant differential operators, and spherical functions, Pure Appl. Math., vol. 113, Academic Press, Inc., Orlando, FL, 1984. MR 0754767 (86c:22017)
  • 11. V. V. Volchkov, On a problem of Zalcman and its generalizations, Mat. Zametki 53 (1993), no. 2, 30-36; English transl., Math. Notes 53 (1993), no. 1-2, 134-138. MR 1220807 (94i:43008)
  • 12. M. El Harchaoui, Inversion de la transformation de Pompeiu dans le disque hyperbolique (cas de deux disques), Publ. Mat. 37 (1993), 133-164. MR 1240929 (94m:43019)
  • 13. -, Inversion de la transformation de Pompeiu locale dans les espaces hyperboliques réel et complexe: cas des deux boules, J. Anal. Math. 67 (1995), 1-37. MR 1383489 (97e:43006)
  • 14. V. V. Volchkov, Two-radii theorems on spaces of constant curvature, Dokl. Akad. Nauk 347 (1996), no. 3, 300-302; English transl., Dokl. Math. 53 (1996), no. 2, 199-201. MR 1393056 (97g:43006)
  • 15. Vit. V. Volchkov, On functions with zero spherical means on quaternion hyperbolic space, Izv. Ross. Akad. Nauk Ser. Mat. 66 (2002), no. 5, 3-32; English transl., Izv. Math. 66 (2002), no. 5, 875-903. MR 1965935 (2004e:43014)
  • 16. A. T. Fomenko, Symplectic geometry: methods and applications, Moskov. Gos. Univ., Moscow, 1988; English transl., Stud. Contemp. Math., vol. 5, Gordon and Breach Science Publishers, New York, 1988. MR 0994805 (90k:58065)
  • 17. W. Rudin, Function theory in the unit ball of $\Bbb C^n$, Grundlehren Math. Wiss., vol. 241, Springer-Verlag, New York-Berlin, 1980. MR 0601594 (82i:32002)
  • 18. C. A. Berenstein and L. Zalcman, Pompeiu's problem on symmetric spaces, Comment. Math. Helv. 55 (1980), 593-621. MR 0604716 (83d:43012)
  • 19. M. Eguchi, M. Hashizume, and K. Okamoto, The Paley-Wiener theorem for distributions on symmetric spaces, Hiroshima Math. J. 3 (1973), 109-120. MR 0338261 (49:3027)
  • 20. V. V. Volchkov, A definitive version of the local two-radius theorem on hyperbolic spaces, Izv. Ross. Akad. Nauk Ser. Mat. 65 (2001), no. 2, 3-26; English transl., Izv. Math. 65 (2001), no. 2, 207-229. MR 1842839 (2002g:43010)
  • 21. Vit. V. Volchkov, Theorems on spherical means on complex hyperbolic spaces, Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Nauki 2000, no. 4, 7-10. (Russian) MR 1799203 (2001k:32011)
  • 22. -, A definitive version of a local two-radii theorem on a quarternion hyperbolic space, Dokl. Akad. Nauk 384 (2002), no. 4, 449-451. (Russian) MR 1929569 (2003i:43011)
  • 23. V. V. Volchkov, A local two-radius theorem on symmetric spaces, Dokl. Akad. Nauk 381 (2001), no. 6, 727-731. (Russian) MR 1892518 (2003b:43007)
  • 24. Vit. V. Volchkov and N. P. Volchkova, Inversion of the local Pompeiu transformation on a quaternionic hyperbolic space, Dokl. Akad. Nauk 379 (2001), no. 5, 587-590. (Russian) MR 1864615 (2002h:43011)
  • 25. C. A. Berenstein and M. Shahshahani, Harmonic analysis and the Pompeiu problem, Amer. J. Math. 105 (1983), 1217-1229. MR 0714774 (85d:32061)
  • 26. M. Shahshahani and A. Sitaram, The Pompeiu problem in exterior domains in symmetric spaces, Integral Geometry (Brunswick, Maine, 1984), Contemp. Math., vol. 63, Amer. Math. Soc., Providence, RI, 1987, pp. 267-277. MR 0876324 (88e:43008)
  • 27. E. M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Math. Ser., No. 32, Princeton Univ. Press, Princeton, NJ, 1971. MR 0304972 (46:4102)
  • 28. A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher transcendental functions. . 2, Robert E. Krieger Publishing Co., Inc., Melbourne, Fla., 1981. MR 0698780 (84h:33001b)
  • 29. N. Ya. Vilenkin, Special functions and theory of group representations, 2nd ed., ``Nauka'', Moscow, 1991; English transl. of 1st ed., Transl. Math. Monogr., vol. 22, Amer. Math. Soc., Providence, RI, 1968. MR 0229863 (37:5429)
  • 30. A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher transcendental functions. . 1, Robert E. Krieger Publishing Co., Inc., Melbourne, Fla., 1981. MR 0698779 (84h:33001a)
  • 31. V. S. Vladimirov, Generalized functions in mathematical physics, ``Nauka'', Moscow, 1976; English transl. of 2nd Russian ed., ``Mir'', Moscow, 1979. MR 0564116 (80j:46062b)

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Additional Information

Vit. V. Volchkov
Affiliation: Donetsk National University, Department of Mathematical Analysis and Function Theory, Ulitsa A. Malyshko 3, Donetsk, 83053, Ukraine
Email: volchkov@univ.donetsk.ua

N. P. Volchkova
Affiliation: Donetsk National University, Department of Mathematical Analysis and Function Theory, Ulitsa A. Malyshko 3, Donetsk, 83053, Ukraine
Email: volchkov@univ.donetsk.ua

DOI: https://doi.org/10.1090/S1061-0022-04-00830-1
Keywords: Quaternionic hyperbolic space, Pompeiu transformation, theorem on two radii
Received by editor(s): October 28, 2002
Published electronically: July 29, 2004
Additional Notes: Partly supported by grant no. 01.07/00241 from the Foundation for Basic Research of Ukraine.
Article copyright: © Copyright 2004 American Mathematical Society

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