Geodesic
Local two-radii theorems on the multi-dimensional sphere
Izvestiya. Mathematics , Tome 78 (2014) no. 1, pp. 1-21

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Consider those functions on the $n$-dimensional sphere that have zero integrals over all geodesic balls with centres in a given set $E$. We obtain a description of such functions in the case when $E$ is a geodesic sphere on $\mathbb S^n$. We also find a criterion for the existence of non-zero functions with this property in the case when the set of centres is the union of two geodesic spheres. We obtain analogues of these results for quasi-analytic classes of functions.
Keywords: two-radii theorems, Legendre functions, spherical harmonics, quasi-analytic classes. 
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V. V. Volchkov; Vit. V. Volchkov. Local two-radii theorems on the multi-dimensional sphere. Izvestiya. Mathematics , Tome 78 (2014) no. 1, pp. 1-21. http://geodesic.mathdoc.fr/item/IM2_2014_78_1_a0/