Comptes Rendus
no. 12
Injectivity of the spherical means operator
[Injectivité de l'operateur de moyen spherique]
Alexander G. Ramm1, 2
1 LMA-CNRS, 31, Chemin J. Aiguier, 13402 Marseille, France
2 Mathematics Department, Kansas State University, Manhattan, KS 66506-2602, USA
Comptes Rendus. Mathématique, Volume 335 (2002) no. 12, pp. 1033-1038.

Soit S une surface de ℝ n qui divise l'espace en deux composantes connectées D1 and D2. Soit f∈C 0 ∞ (ℝ n ) une fonction à valeurs réeles, suppf⊂D1. Considérons

Mf:=m(y,r):=∫ ℝ n f(z)δ(|y-z|-r)dz,
où δ est la delta-fonction, y∈S 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 ℝ n which divides the space into two connected components D1 and D2. Let f∈C 0 (ℝ n ) be some real-valued compactly supported function with suppf⊂D1. Consider

Mf:=m(y,r):=∫ ℝ n 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 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.

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

1 LMA-CNRS, 31, Chemin J. Aiguier, 13402 Marseille, France
2 Mathematics Department, Kansas State University, Manhattan, KS 66506-2602, USA 
@article{CRMATH_2002__335_12_1033_0,
     author = {Alexander G. Ramm},
     title = {Injectivity of the spherical means operator},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1033--1038},
     publisher = {Elsevier},
     volume = {335},
     number = {12},
     year = {2002},
     doi = {10.1016/S1631-073X(02)02608-0},
     language = {en},
}
TY  - JOUR
AU  - Alexander G. Ramm
TI  - Injectivity of the spherical means operator
JO  - Comptes Rendus. Mathématique
PY  - 2002
SP  - 1033
EP  - 1038
VL  - 335
IS  - 12
PB  - Elsevier
DO  - 10.1016/S1631-073X(02)02608-0
LA  - en
ID  - CRMATH_2002__335_12_1033_0
ER  - 
%0 Journal Article
%A Alexander G. Ramm
%T Injectivity of the spherical means operator
%J Comptes Rendus. Mathématique
%D 2002
%P 1033-1038
%V 335
%N 12
%I Elsevier
%R 10.1016/S1631-073X(02)02608-0
%G en
%F CRMATH_2002__335_12_1033_0
Alexander G. Ramm. Injectivity of the spherical means operator. Comptes Rendus. Mathématique, Volume 335 (2002) no. 12, pp. 1033-1038. doi : 10.1016/S1631-073X(02)02608-0. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02608-0/

[1] M. Agranovsky; E. Quinto Injectivity of the spherical mean operator and related problems, Pitman Res. Notes Math. Ser., 347, Longman, Harlow, 1996, pp. 12-36

[2] L. Anderson On the determination of the function from spherical averages, SIAM J. Math. Anal., Volume 19 (1988), pp. 214-232

[3] R. Courant; D. Hilbert Methods of Mathematical Physics, Interscience, New York, 1962

[4] M. Lavrent'ev; V. Romanov; S. Shishatskii Ill-Posed Problems of Mathematical Physics and Analysis, American Mathematical Society, Providence, RI, 1986

[5] P. Morse; H. Feshbach Methods of Theoretical Physics, McGraw-Hill, New York, 1953

[6] A.G. Ramm Scattering by Obstacles, Reidel, Dordrecht, 1986 (pp. 1–442)

[7] A.G. Ramm Multidimensional Inverse Scattering Problems, Longman Scientific & Wiley, New York, 1992 (pp. 1–385; Expanded Russian edition: Mir, Moscow, 1994, pp. 1–496)

[8] A.G. Ramm Spectral properties of the Schrödinger operator in some domains with infinite boundaries, Soviet Math. Dokl., Volume 152 (1963), pp. 282-285

[9] A.G. Ramm Spectral properties of the Schrödinger operator in some infinite domains, Mat. Sb., Volume 66 (1965), pp. 321-343

Cité par Sources :

Commentaires - Politique

Ces articles pourraient vous intéresser

Erratum entitled: Recovery of small electromagnetic inhomogeneities from partial boundary measurements published in Série IIb, t. 330, n∘ 3, pp. 199–205

Habib Ammari; Alexander G. Ramm

C. R. Méca (2002)

Recovery of small electromagnetic inhomogeneities from partial boundary measurements

Habib Ammari; Alexander G. Ramm

C. R. Méca (2002)