Injectivity of the spherical means operator

✍ Scribed by Alexander G Ramm

• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 70 KB
• Volume: 335
• Category: Article
• ISSN: 1631-073X
• DOI: 10.1016/s1631-073x(02)02608-0

✦ Synopsis

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

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.