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), p. 207-239 / Harvested from Numdam

Résumé

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

(*)Lp(Ω,a1)+⋯+Lp(Ω,am)estunsous-espacefermédeLp(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 $\Omega \subset {\bf R}^2$, $1\le p< \infty $, et si la frontière de $\Omega $ est une courbe lipschitzienne (cette dernière condition peut être affaiblie), alors $(*)$ est valable. Pour $\Omega \subset {\bf R}^n$, $n\ge 3$, la situation est différente : $(*)$ n’est pas nécessairement vrai même si $\Omega $ est convexe ayant une frontière lisse. Or $(*)$ est valable si $\Omega \subset {\bf R}^3$ 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 $.

Abstract

For $a\in {\bf R}^n\backslash \lbrace 0\rbrace $ and $\Omega $ an open bounded subset of ${\bf R}^n$ definie $L^p(\Omega ,a)$ as the closed subset of $L^p(\Omega )$ 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 {\bf R}^n\backslash \lbrace 0\rbrace $, $\nu =1,\ldots , m$, we study for which $\Omega $ it is true that the vector space

(*)Lp(Ω,a1)+⋯+Lp(Ω,am)isaclosedsubspaceofLp(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 $\Omega \subset {\bf R}^2$, the boundary of $\Omega $ is a Lipschitz curve (this condition can be relaxes), and $1\le p< \infty $, then $(*)$ holds. For $\Omega \subset {\bf R}^n$, $n\ge 3$, 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 $\Omega \subset {\bf R}^3$ 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 $.

Détail

Publié le : 1984-01-01

MR 85j:44002 | Zbl 0521.46018

DOI : https://doi.org/10.5802/aif.957

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