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

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 34.1 (1984): 207-239. <http://eudml.org/doc/74617>.

@article{Boman1984,
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\begin\{\}(*)\qquad \quad L^p(\Omega ,a^1)+\cdots + L^p(\Omega ,a^m) \text\{is\} \text\{a\} \text\{closed\} \text\{subspace\} \text\{of\} L^p(\Omega ).\end\{\}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&lt; \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 $.},
author = {Boman, Jan},
journal = {Annales de l'institut Fourier},
keywords = {sums of Ridge functions; range of the X-ray transform; image reconstruction from projections; tomography; matrix-valued differential operator; Lipschitz curve; boundary has non-vanishing principal curvatures},
language = {eng},
number = {1},
pages = {207-239},
publisher = {Association des Annales de l'Institut Fourier},
title = {On the closure of spaces of sums of ridge functions and the range of the $X$-ray transform},
url = {http://eudml.org/doc/74617},
volume = {34},
year = {1984},
}

TY - JOUR
AU - Boman, Jan
TI - On the closure of spaces of sums of ridge functions and the range of the $X$-ray transform
JO - Annales de l'institut Fourier
PY - 1984
PB - Association des Annales de l'Institut Fourier
VL - 34
IS - 1
SP - 207
EP - 239
AB - 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\begin{}(*)\qquad \quad L^p(\Omega ,a^1)+\cdots + L^p(\Omega ,a^m) \text{is} \text{a} \text{closed} \text{subspace} \text{of} L^p(\Omega ).\end{}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&lt; \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 $.
LA - eng
KW - sums of Ridge functions; range of the X-ray transform; image reconstruction from projections; tomography; matrix-valued differential operator; Lipschitz curve; boundary has non-vanishing principal curvatures
UR - http://eudml.org/doc/74617
ER -