Injectivity of the spherical means operator

Let S be a surface in ℝ n which divides the space into two connected components D 1 and D 2 . Let f ∈ C 0 ( ℝ n ) be some real-valued compactly supported function with supp f ⊂ D 1 . Consider M f : = m ( y , r ) : = ∫ ℝ n f ( z ) δ ( | y - z | - r ) d z , where δ is the delta-function, y ∈ S and r >0 are arbitrary. A general, local at infinity, condition on S is given, under which M is injective, that is, Mf =0 implies f =0. The injectivity result is extended to the case when the Fourier transform of f is quasianalytic, so that compactness of support of f is not assumed. A sufficient condition on S is given, under which M −1 can be analytically constructed. Two examples of inversion formulas are given: when S is a plane, and when S is a sphere. These formulas can be used in applications.