Geodesic
New theorems on the mean for solutions of the Helmholtz equation
Sbornik. Mathematics, Tome 79 (1994) no. 2, pp. 281-286 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
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     title = {New theorems on the~mean for solutions of {the~Helmholtz} equation},
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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/

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