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St. Petersburg Mathematical Journal

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A local two-radii theorem on the sphere


Author: Vit. V. Volchkov
Translated by: B. M. Bekker
Original publication: Algebra i Analiz, tom 16 (2004), nomer 3.
Journal: St. Petersburg Math. J. 16 (2005), 453-475
MSC (2000): Primary 26B15, 44A15, 49Q15
Published electronically: May 2, 2005
MathSciNet review: 2083565
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Abstract | References | Similar Articles | Additional Information

Abstract: Various classes of functions with vanishing integrals over all balls of a fixed radius on the sphere ${\mathbb S}^n$ are studied. For such functions, uniqueness theorems are proved, and representations in the form of series in special functions are obtained. These results made it possible to completely resolve the problem concerning the existence of a nonzero function with vanishing integrals over all balls on ${\mathbb S}^n$ the radii of which belong to a given two-element set.


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  • 1. Sigurdur Helgason, Groups and geometric analysis, Pure and Applied Mathematics, vol. 113, Academic Press, Inc., Orlando, FL, 1984. Integral geometry, invariant differential operators, and spherical functions. MR 754767
  • 2. K. Berensteĭn and D. Struppa, Complex analysis and convolution equations, Current problems in mathematics. Fundamental directions, Vol. 54 (Russian), Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1989, pp. 5–111 (Russian). MR 1039621
  • 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
  • 4. Ivan Netuka and Jiří 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
  • 5. Lawrence Zalcman, Supplementary bibliography to: “A bibliographic survey of the Pompeiu problem” [in Approximation by solutions of partial differential equations (Hanstholm, 1991), 185–194, Kluwer Acad. Publ., Dordrecht, 1992; MR1168719 (93e:26001)], Radon transforms and tomography (South Hadley, MA, 2000) Contemp. Math., vol. 278, Amer. Math. Soc., Providence, RI, 2001, pp. 69–74. MR 1851479, 10.1090/conm/278/04595
  • 6. L. Tchakaloff, Sur un problème de D. Pompéiu, Annuaire [Godišnik] Univ. Sofia. Fac. Phys.-Math. Livre 1. 40 (1944), 1–14 (Bulgarian, with French summary). MR 0031980
  • 7. Carlos A. Berenstein and Lawrence Zalcman, Pompeiu’s problem on symmetric spaces, Comment. Math. Helv. 55 (1980), no. 4, 593–621. MR 604716, 10.1007/BF02566709
  • 8. Stephen A. Williams, Analyticity of the boundary for Lipschitz domains without the Pompeiu property, Indiana Univ. Math. J. 30 (1981), no. 3, 357–369. MR 611225, 10.1512/iumj.1981.30.30028
  • 9. Carlos A. Berenstein and Mehrdad Shahshahani, Harmonic analysis and the Pompeiu problem, Amer. J. Math. 105 (1983), no. 5, 1217–1229. MR 714774, 10.2307/2374339
  • 10. Peter Ungar, Freak theorem about functions on a sphere, J. London Math. Soc. 29 (1954), 100–103. MR 0057963
  • 11. D. H. Armitage, The Pompeiu property for spherical polygons, Proc. Roy. Irish Acad. Sect. A 96 (1996), no. 1, 25–32. MR 1644624
  • 12. Carlos A. Berenstein and Roger Gay, A local version of the two-circles theorem, Israel J. Math. 55 (1986), no. 3, 267–288. MR 876395, 10.1007/BF02765026
  • 13. Mimoun El Harchaoui, 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 (French, with English summary). MR 1383489, 10.1007/BF02787785
  • 14. V. V. Volchkov, Spherical means theorems on symmetric spaces, Mat. Sb. 192 (2001), no. 9, 17–38 (Russian, with Russian summary); English transl., Sb. Math. 192 (2001), no. 9-10, 1275–1296. MR 1867008, 10.1070/SM2001v192n09ABEH000593
  • 15. V. V. Volchkov, A local two-radius theorem on symmetric spaces, Dokl. Akad. Nauk 381 (2001), no. 6, 727–731 (Russian). MR 1892518
  • 16. Vit. V. Volchkov, On functions with zero spherical means on a quaternion hyperbolic space, Izv. Ross. Akad. Nauk Ser. Mat. 66 (2002), no. 5, 3–32 (Russian, with Russian summary); English transl., Izv. Math. 66 (2002), no. 5, 875–903. MR 1965935, 10.1070/IM2002v066n05ABEH000401
  • 17. B. Ya. Levin, Distribution of zeros of entire functions, Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow, 1956 (Russian). MR 0087740
    B. Ja. Levin, Distribution of zeros of entire functions, Revised edition, Translations of Mathematical Monographs, vol. 5, American Mathematical Society, Providence, R.I., 1980. Translated from the Russian by R. P. Boas, J. M. Danskin, F. M. Goodspeed, J. Korevaar, A. L. Shields and H. P. Thielman. MR 589888
  • 18. Rolf Schneider, Functions on a sphere with vanishing integrals over certain subspheres., J. Math. Anal. Appl. 26 (1969), 381–384. MR 0237723
  • 19. Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton University Press, Princeton, N.J., 1971. Princeton Mathematical Series, No. 32. MR 0304972
  • 20. N. Ya. Vilenkin, Spetsialnye funktsii i teoriya predstavlenii grupp, 2nd ed., “Nauka”, Moscow, 1991 (Russian, with Russian summary). MR 1177172
    N. Ja. Vilenkin, Special functions and the theory of group representations, Translated from the Russian by V. N. Singh. Translations of Mathematical Monographs, Vol. 22, American Mathematical Society, Providence, R. I., 1968. MR 0229863
  • 21. V. V. Volchkov, New mean-value theorems for solutions of the Helmholtz equation, Mat. Sb. 184 (1993), no. 7, 71–78 (Russian, with Russian summary); English transl., Russian Acad. Sci. Sb. Math. 79 (1994), no. 2, 281–286. MR 1235290, 10.1070/SM1994v079n02ABEH003500
  • 22. Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger, and Francesco G. Tricomi, Higher transcendental functions. Vols. I, II, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1953. Based, in part, on notes left by Harry Bateman. MR 0058756
  • 23. È. Ya. \cyr{R}iekstyn′sh and È. Ja. Riekstyn′š, Asimptoticheskie razlozheniya integralov. Tom 1, Izdat. “Zinatne”, Riga, 1974 (Russian). MR 0374775
  • 24. Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger, and Francesco G. Tricomi, Higher transcendental functions. Vols. I, II, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1953. Based, in part, on notes left by Harry Bateman. MR 0058756
  • 25. Fritz John, Plane waves and spherical means applied to partial differential equations, Interscience Publishers, New York-London, 1955. MR 0075429
  • 26. V. V. Volchkov, Uniqueness theorems for multiple lacunary trigonometric series, Mat. Zametki 51 (1992), no. 6, 27–31, 156 (Russian, with Russian summary); English transl., Math. Notes 51 (1992), no. 5-6, 550–552. MR 1187473, 10.1007/BF01263296
  • 27. A. N. Kolmogorov and S. V. Fomin, Elementy teorii funktsii i funktsionalnogo analiza, 6th ed., “Nauka”, Moscow, 1989 (Russian). With a supplement, “Banach algebras”, by V. M. Tikhomirov. MR 1025126
    A. N. Kolmogorov and S. V. Fomin, Elements of the theory of functions and functional analysis. Vol. 1. Metric and normed spaces, Graylock Press, Rochester, N. Y., 1957. Translated from the first Russian edition by Leo F. Boron. MR 0085462
    A. N. Kolmogorov and S. V. Fomin, Elements of the theory of functions and functional analysis. Vol. 2: Measure. The Lebesgue integral. Hilbert space, Translated from the first (1960) Russian ed. by Hyman Kamel and Horace Komm, Graylock Press, Albany, N.Y., 1961. MR 0118796

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Additional Information

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

DOI: https://doi.org/10.1090/S1061-0022-05-00861-7
Keywords: Spherical harmonics, Legendre functions, two-radii theorem, Pompeiu transformation
Received by editor(s): June 2, 2003
Published electronically: May 2, 2005
Additional Notes: Supported by the Ukraine Foundation for fundamental research (project no. 01.07/00241).
Article copyright: © Copyright 2005 American Mathematical Society