FPM 1996, vol. 2, no. 4, pp. 1195-1204

FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

1996, VOLUME 2, NUMBER 4, PAGES 1195-1204

Harmonic solution for the inverse problem of the Newtonian potential theory

J. Bosgiraud

Abstract

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We study from a theoretical point of view the Backus and Gilbert method in the case of Newtonian potential. If a mass distribution $m$ on a open set $Ω$ creates a Newtonian potential $U$ ^m, which is known on an infinity of points $(y$ _n)_{n Î N} out of $$\overline{\Omega}$$ , we characterize the solution $m$ ₀, obtained as a generalization of the Backus and Gilbert method, as the projection of $m$ (for the scalar product of $L$ ₂(Ω)) on a subspace of harmonic functions; this subspace may be the subspace of all harmonic, square-integrable functions (for example, if $Ω$ is a starlike domain). Then we study the reproducing kernel $B$ associated to this projection, which satisfies

m

₀(x) = ∫_Ω B(x,y)m(y) dy

for any $m$ Î L₂(Ω).

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