TITLE: Vanishing of integral on hemispheres implies vanishing of function?
QUESTION [6 upvotes]: Consider a function $F$ on the half space $\{(x,y,z)|z>0\}$. If $F$ is analytic, it is straightforward to show that 
A) The integral of $F$ over the hemisphere $(x-x_0)^2 + (y-y_0)^2 + z^2 = R^2$ vanishes for all $x_0$, $y_0$, and $R$.
implies
B) $F = 0$ everywhere
My question is whether this is known to be true for any continuous function (or under some less restrictive conditions).

REPLY [3 votes]: As @alvarezpaiva notes, this is a question about hyperbolic hyperplane (Radon) transform. This has been studied. See, e.g., Kurusa, the Radon transform in hyperbolic space [Geometriae Dedicata, 1991] (available for free, thanks to Springer). and references therein (Kurusa inverts it on fairly natural subspaces of $L^2,$ I am not sure if his results are optimal.