The range of the spherical mean value operator for functions supported in a ball

Suppose n > 1 is an odd integer, f is a smooth function supported in a ball B with boundary S, and u is the solution of the initial value problem utt - Delta(x)u = 0, (x,t) is an element of R-n x [0,infinity); u(x, t = 0) = 0, u(t)(x,t = 0) = f (x), x is an element of R-n. We characterize the range of the map f -> u\S x[0,infinity) and give a stable scheme for the inversion of this map. This also characterizes the range of the map sending f to its mean values over spheres centred on S.