  We study a $(k+1)$-dimensional hyperbolic space of a negative constant
sectional curvature $\kappa=-1/\rho^2$. Let $\lambda$ be a real eigenvalue and
$f_{\lambda} (x)$ be an eigenfunction of the hyperbolic Laplacian assuming a
non-zero value at $x_0$. Then the average value of $f_{\lambda}(x)$ over any
sphere centered at $x_0$ allows to identify the corresponding eigenvalue
$\lambda$ uniquely as long as that average value is large enough. Otherwise, to
identify the corresponding eigenvalue uniquely, we need to make sure that the
computed average value is not zero and then we need to compute an additional
average value of $f_{\lambda}(x)$ over a small enough sphere centered at the
same point $x_0$.
