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 is an element of C-0(R-n) be some real-valued compactly supported function with supp f subset of D-1. Consider Mf := m(y, r) := integral(R)(n) f (z)delta(\y - z\ - r) dz, where delta is the delta-function, y is an element of 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. (C) 2002 Academie des science/Editions scientifiques et medicales Elsevier SAS.