Injectivity of the spherical means operator

Let S be a surface in R which divides the space into two connected components D1 and D2. Let f ∈ C0(R) be some real-valued compactly supported function with supp f ⊂ D1. Consider Mf := m(y, r) := ∫ Rn f(z)δ(|y − z| − r)dz, 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 whichM−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 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS Injectivité de l’operateur de moyen spherique Résumé. Soit S une surface de R qui divise l’espace en deux composantes connectées D1 and D2. Soit f ∈ C∞ 0 (R) une fonction à valeurs réeles, supp f ⊂ D1. Considérons