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 −4xu = 0, (x, t) ∈ R × [0,∞); u(x, t=0) = 0, ut(x, t=0) = f(x), x ∈ R. We characterize the range of the map f 7→ u|S×[0,∞) and give a stable scheme for the inversion of this map. This also characterizes the… CONTINUE READING