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 $\Omega$ creates a Newtonian potential $Um$, which is known on an infinity of points $\left(y$n)n Î N out of $\overline\left\{\Omega\right\}$, 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 $\Omega$ 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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