Conic injectivity sets for the Radon transformation on spheres
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V. V. Volchkov and Vit. V. Volchkov
Translated by: A. Plotkin - St. Petersburg Math. J. 27 (2016), 709-730
- DOI: https://doi.org/10.1090/spmj/1413
- Published electronically: July 26, 2016
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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.References
- A. B. Aleksandrov, $A$-integrability of boundary values of harmonic functions, Mat. Zametki 30 (1981), no. 1, 59–72, 154 (Russian). MR 627941
- Sigurdur Helgason, Groups and geometric analysis, Mathematical Surveys and Monographs, vol. 83, American Mathematical Society, Providence, RI, 2000. Integral geometry, invariant differential operators, and spherical functions; Corrected reprint of the 1984 original. MR 1790156, DOI 10.1090/surv/083
- V. V. Volchkov, Integral geometry and convolution equations, Kluwer Academic Publishers, Dordrecht, 2003. MR 2016409, DOI 10.1007/978-94-010-0023-9
- Fritz John, Plane waves and spherical means applied to partial differential equations, Interscience Publishers, New York-London, 1955. MR 0075429
- R. Courant and D. Hilbert, Methods of mathematical physics. Vol. II: Partial differential equations, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. (Vol. II by R. Courant.). MR 0140802
- V. V. Volchkov, Injectivity sets for the Radon transform on spheres, Izv. Ross. Akad. Nauk Ser. Mat. 63 (1999), no. 3, 63–76 (Russian, with Russian summary); English transl., Izv. Math. 63 (1999), no. 3, 481–493. MR 1712132, DOI 10.1070/im1999v063n03ABEH000248
- Mark Agranovsky, Carlos Berenstein, and Peter Kuchment, Approximation by spherical waves in $L^p$-spaces, J. Geom. Anal. 6 (1996), no. 3, 365–383 (1997). MR 1471897, DOI 10.1007/BF02921656
- Mark L. Agranovsky and Eric Todd 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, DOI 10.1006/jfan.1996.0090
- M. L. Agranovsky, V. V. Volchkov, and L. A. Zalcman, Conical uniqueness sets for the spherical Radon transform, Bull. London Math. Soc. 31 (1999), no. 2, 231–236. MR 1664137, DOI 10.1112/S0024609398005396
- 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
- V. P. Burskiĭ, Investigation methods of boundary value problems for general differential equations, Kiev, Naukova Dumka, 2002. (Russian)
- Vit. V. Volchkov, A local two-radius theorem on the sphere, Algebra i Analiz 16 (2004), no. 3, 24–55 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 16 (2005), no. 3, 453–475. MR 2083565, DOI 10.1090/S1061-0022-05-00861-7
- Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, N.J., 1971. MR 0304972
- Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger, and Francesco G. Tricomi, Higher transcendental functions. Vols. I, II, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1953. Based, in part, on notes left by Harry Bateman. MR 0058756
- Sigurdur Helgason, Integral geometry and Radon transforms, Springer, New York, 2011. MR 2743116, DOI 10.1007/978-1-4419-6055-9
- Valery V. Volchkov and Vitaly V. Volchkov, Harmonic analysis of mean periodic functions on symmetric spaces and the Heisenberg group, Springer Monographs in Mathematics, Springer-Verlag London, Ltd., London, 2009. MR 2527108, DOI 10.1007/978-1-84882-533-8
- Serge Lang, $\textrm {SL}_{2}(\textbf {R})$, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1975. MR 0430163
- N. Ja. Vilenkin, Special functions and the theory of group representations, Translations of Mathematical Monographs, Vol. 22, American Mathematical Society, Providence, R.I., 1968. Translated from the Russian by V. N. Singh. MR 0229863
- 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
- Lars Hörmander, The analysis of linear partial differential operators. I, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 256, Springer-Verlag, Berlin, 1990. Distribution theory and Fourier analysis. MR 1065993, DOI 10.1007/978-3-642-61497-2
- Ya. B. Lopatinskiĭ, Vvedenie v sovremennuyu teoriyu differentsial′nykh uravneniĭ v chastnykh proizvodnykh, “Naukova Dumka”, Kiev, 1980 (Russian). MR 591676
Bibliographic Information
- V. V. Volchkov
- Affiliation: Donetsk national university, 24 Universitetskaya str., Donetsk 83001, Ukraine
- Email: valeriyvolchkov@gmail.com
- Received by editor(s): December 16, 2014
- Published electronically: July 26, 2016
- © Copyright 2016 American Mathematical Society
- Journal: St. Petersburg Math. J. 27 (2016), 709-730
- MSC (2010): Primary 44A12
- DOI: https://doi.org/10.1090/spmj/1413
- MathSciNet review: 3582940