Geodesic
New theorems on the~mean for solutions of the~Helmholtz equation
Sbornik. Mathematics, Tome 79 (1994) no. 2, pp. 281-286

Voir la notice de l'article provenant de la source Math-Net.Ru

It is proved that the solutions of the equation $\Delta u+u=0$ are characterized by vanishing of integrals over all balls in $R^n$ with radii belonging to the zero set of the Bessel function $J_{n/2}$. This result enables us to get a solution of the Pompeiu problem on the class of functions of slow growth in terms of approximation in $L(R^n)$ by linear combinations with special radii. 
@article{SM_1994_79_2_a2,
     author = {V. V. Volchkov},
     title = {New theorems on the~mean for solutions of {the~Helmholtz} equation},
     journal = {Sbornik. Mathematics},
     pages = {281--286},
     publisher = {mathdoc},
     volume = {79},
     number = {2},
     year = {1994},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1994_79_2_a2/}
}
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V. V. Volchkov. New theorems on the~mean for solutions of the~Helmholtz equation. Sbornik. Mathematics, Tome 79 (1994) no. 2, pp. 281-286. http://geodesic.mathdoc.fr/item/SM_1994_79_2_a2/