Geodesic
The extension problem for functions with zero weighted spherical means
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 8 (2017), pp. 17-26.

Voir la notice de l'article provenant de la source Math-Net.Ru

We study functions on a sphere with pricked point having zero integrals with a given weight over all admissible “hemispheres”. We find a condition under which a point is the removable set for such class of functions. We show that this condition cannot be dropped or substantially weakened.
Keywords: spherical harmonics, generalized Funk transform, mean periodicity. 
@article{IVM_2017_8_a1,
     author = {Vit. V. Volchkov and N. P. Volchkova},
     title = {The extension problem for functions with zero weighted spherical means},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {17--26},
     publisher = {mathdoc},
     number = {8},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2017_8_a1/}
}
TY  - JOUR
AU  - Vit. V. Volchkov
AU  - N. P. Volchkova
TI  - The extension problem for functions with zero weighted spherical means
JO  - Izvestiâ vysših učebnyh zavedenij. Matematika
PY  - 2017
SP  - 17
EP  - 26
IS  - 8
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IVM_2017_8_a1/
LA  - ru
ID  - IVM_2017_8_a1
ER  - 
%0 Journal Article
%A Vit. V. Volchkov
%A N. P. Volchkova
%T The extension problem for functions with zero weighted spherical means
%J Izvestiâ vysših učebnyh zavedenij. Matematika
%D 2017
%P 17-26
%N 8
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IVM_2017_8_a1/
%G ru
%F IVM_2017_8_a1
Vit. V. Volchkov; N. P. Volchkova. The extension problem for functions with zero weighted spherical means. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 8 (2017), pp. 17-26. http://geodesic.mathdoc.fr/item/IVM_2017_8_a1/

[1] Shabat B. V., Vvedenie v kompleksnyi analiz, v. 2, Nauka, M., 1985

[2] Martio O., Ryazanov V., Srebro U., Yakubov E., Moduli in modern mapping theory, Springer, New York, 2009 | MR | Zbl

[3] Axler S., Bourdon P., Ramey W., Harmonic function theory, Springer-Verlag, New York, 1992 | MR | Zbl

[4] Markushevich A. I., Izbrannye glavy teorii analiticheskikh funktsii, Nauka, M., 1976 | MR

[5] Trokhimchuk Yu. Yu., Nepreryvnye otobrazheniya i usloviya monogennosti, Fizmatgiz, M., 1963

[6] Helgason S., Integral geometry and Radon transforms, Springer, New York, 2010 | MR

[7] Volchkov V. V., Integral geometry and convolution equations, Kluwer Academic Publ., Dordrecht, 2003 | MR | Zbl

[8] Volchkov V. V., Volchkov Vit. V., Harmonic analysis of mean periodic functions on symmetric spaces and the Heisenberg group, Springer, London, 2009 | MR | Zbl

[9] Volchkov V. V., Volchkov Vit. V., Offbeat integral geometry on symmetric spaces, Birkhäuser, Basel, 2013 | MR | Zbl

[10] Quinto E. T., “Pompeiu transforms on geodesic spheres in real analytic manifolds”, Israel J. Math., 84 (1983), 353–363 | DOI | MR

[11] Quinto E. T., “Radon transforms on curves in the plane”, Tomography, Impedance Imaging, and Integral Geometry (South Hadley, MA), Lectures in Appl. Math., 30, eds. E. T. Quinto, M. Cheney, P. Kuchment, 1994, 231–244 | MR | Zbl

[12] Zhou Y., “Two radius support theorem for the sphere transform”, J. Math. Anal. Appl., 254 (2001), 120–137 | DOI | MR | Zbl

[13] Volchkov V. V., “Reshenie problemy nositelya dlya nekotorykh klassov funktsii”, Matem. sb., 188:9 (1997), 13–30 | DOI | Zbl

[14] Stein I., Veis G., Vvedenie v garmonicheskii analiz na evklidovykh prostranstvakh, Mir, M., 1974

[15] Funk P., “Über eine geometrische Anwendung der Abelschen Integralgleichung”, Math. Annal., 77 (1916), 129–135 | DOI | MR

[16] Rubin B., “Inversion and characterization of the hemispherical transform”, J. D'Analyse Math., 77 (1999), 105–128 | DOI | MR | Zbl

[17] Campi S., “On the reconstruction of a star-shaped body from its “half-volumes””, J. Austral. Math. Soc. (Ser. A), 37 (1984), 243–257 | DOI | MR | Zbl

[18] Beitmen G., Erdeii A., Vysshie tranctsendentnye funktsii, v. 1, Nauka, M., 1973 | MR

[19] Volchkov Vit. V., Savostyanova I. M., “O yadre polusfericheskogo preobrazovaniya Funka i ego lokalnykh analogakh”, Ukr. matem. vestn., 10:4 (2013), 575–594