On the closure of spaces of sums of ridge functions and the range of the $X$-ray transform

Boman, Jan

Annales de l'Institut Fourier, Tome 34 (1984) no. 1, pp. 207-239.

Soient $a\in {\mathbf{R}}^n\setminus \left\{0\right\}$ et $\Omega$ un ouvert borné de ${\mathbf{R}}^n$. Soit $L^p(\Omega ,a)$ le sous-espace fermé de $L^p\left(\Omega \right)$ formé des fonctions constantes presque partout sur presque toutes les lignes parallèles à $a$. Pour un ensemble donné de directions $a^{\nu }\in {\mathbf{R}}^n\setminus \left\{0\right\}$, $\nu =1,...,m$, on veut déterminer les $\Omega$ pour lesquels

$$(*)L^p(\Omega ,a^1)+\cdots +L^p(\Omega ,a^m)\text{est}\text{un}\text{sous-espace}\text{fermé}\text{de}{L}^p\left(\Omega \right).$$

On rencontre ce problème dans l’étude des reconstructions des images à partir des projections (tomographie). C’est un problème essentiellement équivalent que de décider si un certain opérateur à valeurs matricielles a son image fermée. Si $\Omega \subset {\mathbf{R}}^2$, $1\le p<\infty$, et si la frontière de $\Omega$ est une courbe lipschitzienne (cette dernière condition peut être affaiblie), alors $(*)$ est valable. Pour $\Omega \subset {\mathbf{R}}^n$, $n\ge 3$, la situation est différente : $(*)$ n’est pas nécessairement vrai même si $\Omega$ est convexe ayant une frontière lisse. Or $(*)$ est valable si $\Omega \subset {\mathbf{R}}^3$ est convexe et si en outre les courbures principales de la frontière sont non-nulles en un nombre fini de points déterminés par les $a^{\nu }$.

For $a\in {\mathbf{R}}^n\setminus \left\{0\right\}$ and $\Omega$ an open bounded subset of ${\mathbf{R}}^n$ definie $L^p(\Omega ,a)$ as the closed subset of $L^p\left(\Omega \right)$ consisting of all functions that are constant almost everywhere on almost all lines parallel to $a$. For a given set of directions $a^{\nu }\in {\mathbf{R}}^n\setminus \left\{0\right\}$, $\nu =1,...,m$, we study for which $\Omega$ it is true that the vector space

$$(*)L^p(\Omega ,a^1)+\cdots +L^p(\Omega ,a^m)\text{is}\text{a}\text{closed}\text{subspace}\text{of}{L}^p\left(\Omega \right).$$

This problem arizes naturally in the study of image reconstruction from projections (tomography). An essentially equivalent problem is to decide whether a certain matrix-valued differential operator has closed range. If $\Omega \subset {\mathbf{R}}^2$, the boundary of $\Omega$ is a Lipschitz curve (this condition can be relaxes), and $1\le p<\infty$, then $(*)$ holds. For $\Omega \subset {\mathbf{R}}^n$, $n\ge 3$, the situation is different: $(*)$ is not necessarily true even if $\Omega$ is convex and has smooth boundary. On the other hand we prove that $(*)$ holds if $\Omega \subset {\mathbf{R}}^3$ is convex and the boundary has non-vanishing principal curvatures at a certain finite set of points, which is determined by the set of directions $a^{\nu }$.

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