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

Annales de l'institut Fourier (1984)

- Volume: 34, Issue: 1, page 207-239
- ISSN: 0373-0956

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topBoman, 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< \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< \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 -

## References

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- [9] K.T. SMITH, D.C. SOLMON, S.L. WAGNER, Practical and mathematical aspects of the problem of reconstructing objects from radiographs, Bull. A.M.S., 83 (1977), 1227-1270. Zbl0521.65090MR58 #9394a
- [10] L. SVENSSON, When is the sum of closed subspaces closed? An example arising in computerized tomography, Research Report, Royal Inst. Technology (Stockholm), 1980.
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