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Functions with ball mean values equal to zero on compact two-point homogeneous spaces. (English. Russian original) Zbl 1156.43005
Sb. Math. 198, No. 4, 465-490 (2007); translation from Mat. Sb. 198, No. 4, 21-46 (2007).
In the present paper, the author solves the following Pompeiu type problem: in a compact two point homogeneous space describe the class of functions with zero integral over each ball of prescribed radius, lying in a fixed ball. The author successfully uses a method which he has previously developed for a related problem, namely the analysis of spherically symmetric uniqueness sets for solutions of convolution equations on the Heisenberg group and on symmetric spaces [Vit. V. Volchkov, Dopov. Nats. Akad. Nauk Ukr., Mat. Pryr. Tekh. Nauky 2005, No. 3, 7–11 (2005; Zbl 1088.43003)]. This method is based on the representation of spherical functions as Euclidean Fourier transforms of compactly supported integrable functions and on the reduction of the original equation to a convolution equation on $$R^1$$.

##### MSC:
 43A85 Harmonic analysis on homogeneous spaces 53C65 Integral geometry 26B15 Integration of real functions of several variables: length, area, volume 43A90 Harmonic analysis and spherical functions 53C30 Differential geometry of homogeneous manifolds
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