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, p. 207-239

For aR 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 R n {0}, ν=1,...,m, we study for which Ω 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 ΩR 2 , the boundary of Ω is a Lipschitz curve (this condition can be relaxes), and 1p<, then (*) holds. For ΩR n , n3, 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 ΩR 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 .

Soient aR 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 R n {0}, ν=1,...,m, on veut déterminer les Ω 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 ΩR 2 , 1p<, et si la frontière de Ω est une courbe lipschitzienne (cette dernière condition peut être affaiblie), alors (*) est valable. Pour ΩR n , n3, la situation est différente : (*) n’est pas nécessairement vrai même si Ω est convexe ayant une frontière lisse. Or (*) est valable si ΩR 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 .

@article{AIF_1984__34_1_207_0,
     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},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {34},
     number = {1},
     year = {1984},
     pages = {207-239},
     doi = {10.5802/aif.957},
     zbl = {0521.46018},
     mrnumber = {85j:44002},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1984__34_1_207_0}
}
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. http://www.numdam.org/item/AIF_1984__34_1_207_0/

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