Injectivity of the spherical means operator

Alexander G. Ramm

doi:10.1016/S1631-073X(02)02608-0

Injectivity of the spherical means operator
[Injectivité de l'operateur de moyen spherique]

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

Comptes Rendus. Mathématique, Volume 335 (2002) no. 12, pp. 1033-1038.

Résumé
Abstract

Soit S une surface de $ℝ^{n}$ qui divise l'espace en deux composantes connectées D₁ and D₂. Soit $f \in C_{0}^{\infty} (ℝ^{n})$ une fonction à valeurs réeles, suppf⊂D₁. Considérons

M f : = m (y, r) : = \int_{ℝ^{n}} f (z) δ (| y - z | - r) d z,

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 $f \in C_{0} (ℝ^{n})$ be some real-valued compactly supported function with suppf⊂D₁. Consider

M f : = m (y, r) : = \int_{ℝ^{n}} f (z) δ (| y - z | - r) d z,

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 : 2002-10-04
Accepté le : 2002-10-04
Publié le : 2002-11-20

DOI : 10.1016/S1631-073X(02)02608-0

Affiliations des auteurs :

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

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     title = {Injectivity of the spherical means operator},
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     pages = {1033--1038},
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     volume = {335},
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     doi = {10.1016/S1631-073X(02)02608-0},
     language = {en},
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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/

Bibliographie
Cité par

[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

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E. T. Quinto Support theorems for the spherical Radon transform on manifolds, International Mathematics Research Notices (2006) | DOI:10.1155/imrn/2006/67205
Thomas Schuster; Eric Todd Quinto On a Regularization Scheme for Linear Operators in Distribution Spaces with an Application to the Spherical Radon Transform, SIAM Journal on Applied Mathematics, Volume 65 (2005) no. 4, p. 1369 | DOI:10.1137/s003613990343879x
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