Geodesic
On a problem of Berenstein--Gay and its generalizations
Izvestiya. Mathematics , Tome 74 (2010) no. 4, pp. 691-721

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We obtain a solution of the Berenstein–Gay problem on the local analogue of spectral analysis on Riemannian symmetric spaces $X$ of rank 1. The proof is based on constructing transmutation maps connected with eigenfunction expansions of the Laplace–Beltrami operator on $X$.
Keywords: spectral analysis, functions periodic on average, symmetric spaces. 
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     title = {On a problem of {Berenstein--Gay} and its generalizations},
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V. V. Volchkov; Vit. V. Volchkov. On a problem of Berenstein--Gay and its generalizations. Izvestiya. Mathematics , Tome 74 (2010) no. 4, pp. 691-721. http://geodesic.mathdoc.fr/item/IM2_2010_74_4_a2/