# ANNALES DE L'INSTITUT FOURIER

On the closure of spaces of sums of ridge functions and the range of the $X$-ray transform
Annales de l'Institut Fourier, Volume 34 (1984) no. 1, pp. 207-239.

For $a\in {\mathbf{R}}^{n}\setminus \left\{0\right\}$ and $\Omega$ an open bounded subset of ${\mathbf{R}}^{n}$ definie ${L}^{p}\left(\Omega ,a\right)$ 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

 $\left(*\right)\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}{L}^{p}\left(\Omega ,{a}^{1}\right)+\cdots +{L}^{p}\left(\Omega ,{a}^{m}\right)\phantom{\rule{4pt}{0ex}}\text{is}\phantom{\rule{4pt}{0ex}}\text{a}\phantom{\rule{4pt}{0ex}}\text{closed}\phantom{\rule{4pt}{0ex}}\text{subspace}\phantom{\rule{4pt}{0ex}}\text{of}\phantom{\rule{4pt}{0ex}}{L}^{p}\left(\Omega \right).$

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 $\left(*\right)$ holds. For $\Omega \subset {\mathbf{R}}^{n}$, $n\ge 3$, the situation is different: $\left(*\right)$ is not necessarily true even if $\Omega$ is convex and has smooth boundary. On the other hand we prove that $\left(*\right)$ 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 ${a}^{\nu }$.

Soient $a\in {\mathbf{R}}^{n}\setminus \left\{0\right\}$ et $\Omega$ un ouvert borné de ${\mathbf{R}}^{n}$. Soit ${L}^{p}\left(\Omega ,a\right)$ 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

 $\left(*\right)\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}{L}^{p}\left(\Omega ,{a}^{1}\right)+\cdots +{L}^{p}\left(\Omega ,{a}^{m}\right)\phantom{\rule{4pt}{0ex}}\text{est}\phantom{\rule{4pt}{0ex}}\text{un}\phantom{\rule{4pt}{0ex}}\text{sous-espace}\phantom{\rule{4pt}{0ex}}\text{fermé}\phantom{\rule{4pt}{0ex}}\text{de}\phantom{\rule{4pt}{0ex}}{L}^{p}\left(\Omega \right).$

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 $\left(*\right)$ est valable. Pour $\Omega \subset {\mathbf{R}}^{n}$, $n\ge 3$, la situation est différente : $\left(*\right)$ n’est pas nécessairement vrai même si $\Omega$ est convexe ayant une frontière lisse. Or $\left(*\right)$ 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 ${a}^{\nu }$.

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author = {Boman, Jan},
title = {On the closure of spaces of sums of ridge functions and the range of the $X$-ray transform},
journal = {Annales de l'Institut Fourier},
pages = {207--239},
publisher = {Institut Fourier},
volume = {34},
number = {1},
year = {1984},
doi = {10.5802/aif.957},
zbl = {0521.46018},
mrnumber = {85j:44002},
language = {en},
url = {https://aif.centre-mersenne.org/articles/10.5802/aif.957/}
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Boman, Jan. On the closure of spaces of sums of ridge functions and the range of the $X$-ray transform. Annales de l'Institut Fourier, Volume 34 (1984) no. 1, pp. 207-239. doi : 10.5802/aif.957. https://aif.centre-mersenne.org/articles/10.5802/aif.957/

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