  Let $Z_{r,R}$ be the class of all continuous functions $f$ on the annulus
$\Ann(r,R)$ in the real hyperbolic space $\mathbb B^n$ with spherical means
$M_sf(x)=0$, whenever $s>0$ and $x\in \mathbb B^n$ are such that the sphere
$S_s(x)\subset \Ann(r, R) $ and $B_r(o)\subseteq B_s(x).$ In this article, we
give a characterization for functions in $Z_{r,R}$. In the case $R=\infty$,
this result gives a new proof of Helgason's support theorem for spherical means
in the real hyperbolic spaces.
