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St. Petersburg Mathematical Journal
St. Petersburg Mathematical Journal
ISSN 1547-7371(online) ISSN 1061-0022(print)

 

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: 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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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: http://dx.doi.org/10.1090/S1061-0022-05-00861-7
PII: S 1061-0022(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