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

Boman, Jan

doi:10.5802/aif.957

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.

Résumé
Abstract

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

(*) L^{p} (Ω, a^{1}) + \dots + L^{p} (Ω, a^{m}) est un sous-espace fermé de L^{p} (Ω) .

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 $Ω \subset R^{2}$ , $1 \leq p < \infty$ , et si la frontière de $Ω$ est une courbe lipschitzienne (cette dernière condition peut être affaiblie), alors $(*)$ est valable. Pour $Ω \subset R^{n}$ , $n \geq 3$ , la situation est différente : $(*)$ n’est pas nécessairement vrai même si $Ω$ est convexe ayant une frontière lisse. Or $(*)$ est valable si $Ω \subset R^{3}$ est convexe et si en outre les courbures principales de la frontière sont non-nulles en un nombre fini de points déterminés par les $a^{ν}$ .

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

(*) L^{p} (Ω, a^{1}) + \dots + L^{p} (Ω, a^{m}) is a closed subspace of L^{p} (Ω) .

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 $Ω \subset R^{2}$ , the boundary of $Ω$ is a Lipschitz curve (this condition can be relaxes), and $1 \leq p < \infty$ , then $(*)$ holds. For $Ω \subset R^{n}$ , $n \geq 3$ , the situation is different: $(*)$ is not necessarily true even if $Ω$ is convex and has smooth boundary. On the other hand we prove that $(*)$ holds if $Ω \subset R^{3}$ 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 $a^{ν}$ .

MR Zbl

DOI : 10.5802/aif.957

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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 = {Imprimerie Louis-Jean},
     address = {Gap},
     volume = {34},
     number = {1},
     year = {1984},
     doi = {10.5802/aif.957},
     mrnumber = {85j:44002},
     zbl = {0521.46018},
     language = {en},
     url = {https://www.numdam.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, Tome 34 (1984) no. 1, pp. 207-239. doi : 10.5802/aif.957. https://www.numdam.org/articles/10.5802/aif.957/

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