Injectivity of the spherical means operator
[Injectivité de l'operateur de moyen spherique]
Comptes Rendus. Mathématique, Tome 335 (2002) no. 12, pp. 1033-1038.

Soit S une surface de n qui divise l'espace en deux composantes connectées D1 and D2. Soit fC 0 ( n ) une fonction à valeurs réeles, suppfD1. Considérons

Mf:=m(y,r):= n f(z)δ(|y-z|-r)dz,
δ est la delta-fonction, yS et r>0 sont quelconques. Une condition générale, locale à l'infini, est donnée sur S, sous laquelle M est injective, c.a.d., Mf=0⇒f=0. Le résultat d'injectivité est généralisé dans le cas où la transformée de Fourier de f est quasi-analytique, de façon à ne pas supposer que f est à support compact. Une condition suffisante sur S est donnée sous laquelle M−1 peut être construit analytiquement. Deux exemples de formules d'inversion sont donnés : dans le cas où S est plan, et dans le cas où S est une sphère. Ces formules peuvent etre utilisés dans les applications.

Let S be a surface in n which divides the space into two connected components D1 and D2. Let fC 0 ( n ) be some real-valued compactly supported function with suppfD1. Consider

Mf:=m(y,r):= n f(z)δ(|y-z|-r)dz,
where δ is the delta-function, yS 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.

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DOI : 10.1016/S1631-073X(02)02608-0
Ramm, Alexander G. 1, 2

1 LMA-CNRS, 31, Chemin J. Aiguier, 13402 Marseille, France
2 Mathematics Department, Kansas State University, Manhattan, KS 66506-2602, USA
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Ramm, Alexander G. Injectivity of the spherical means operator. Comptes Rendus. Mathématique, Tome 335 (2002) no. 12, pp. 1033-1038. doi : 10.1016/S1631-073X(02)02608-0. http://archive.numdam.org/articles/10.1016/S1631-073X(02)02608-0/

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