On domain
C
∞
(
R
n
)
{C^\infty }\,({R^n})
we invert the Radon transform that maps a function to its mean values on spheres containing the origin. Our inversion formula implies that if
f
∈
C
∞
(
R
n
)
f\, \in \,{C^\infty }\,({R^n})
and its transform is zero on spheres inside a disc centered at 0, then f is zero inside that disc. We give functions
f
∉
C
∞
(
R
n
)
f\, \notin \,{C^\infty }\,({R^n})
whose transforms are identically zero and we give a necessary condition for a function to be the transform of a rapidly decreasing function. We show that every entire function is the transform of a real analytic function. These results imply that smooth solutions to the classical Darboux equation are determined by the data on any characteristic cone with vertex on the initial surface; if the data is zero near the vertex then so is the solution. If the data is entire then a real analytic solution with that data exists.