Geodesic
Description of the kernel of the generalized Minkowski transform on the sphere
Izvestiya. Mathematics , Tome 79 (2015) no. 1, pp. 40-59

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We study the generalized Minkowski transform that sends functions on a sphere to their weighted integrals over closed geodesics. We solve the problem of describing the kernel of this transform and some related classes of functions. As an application, we obtain new and definitive uniqueness theorems for functions on a sphere with zero weighted averages over great circles.
Keywords: spherical means, Minkowski transform, Legendre functions. 
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V. V. Volchkov; Vit. V. Volchkov; I. M. Savost'yanova. Description of the kernel of the generalized Minkowski transform on the sphere. Izvestiya. Mathematics , Tome 79 (2015) no. 1, pp. 40-59. http://geodesic.mathdoc.fr/item/IM2_2015_79_1_a2/