The relation between the dual and the adjoint Radon transforms

Cnops, J.. "The relation between the dual and the adjoint Radon transforms." Proceedings of the Winter School "Geometry and Physics". Palermo: Circolo Matematico di Palermo, 1991. [135]-142. <http://eudml.org/doc/220237>.

@inProceedings{Cnops1991,
abstract = {[For the entire collection see Zbl 0742.00067.]Let $P^m$ be the set of hyperplanes $\sigma :\langle \vec\{x\},\vec\{\theta \}\rangle =p$ in $\mathcal \{R\}^m$, $S^\{m-1\}$ the unit sphere of $\mathcal \{R\}^m$, $E^m$ the exterior of the unit ball, $T^m$ the set of hyperplanes not passing through the unit ball, $Rf(\vec\{\theta \},p)=\int _\sigma f(\vec\{x\})d\vec\{x\}$ the Radon transform, $R^\#g(\vec\{x\})=\int _\{S^\{m-1\}\}g(\vec\{\theta \},\langle \vec\{x\},\vec\{\theta \}\rangle )dS_\{\vec\{\theta \}\}$ its dual. $R$ as operator from $L^2(\mathcal \{R\}^m)$ to $L^2(S^\{m-1)\}\times \mathcal \{R\})$ is a closable, densely defined operator, $R^*$ denotes the operator given by $(R^*g)(\vec\{x\})=R^\#g(\vec\{x\})$ if the integral exists for $\vec\{x\}\in \mathcal \{R\}^m$ a.e. Then the closure of $R^*$ is the adjoint of $R$. The author shows that the Radon transform and its dual can be linked by two operators of geometrical nature. Using the relation between the dual and the adjoint transform he obtains results regard!},
author = {Cnops, J.},
booktitle = {Proceedings of the Winter School "Geometry and Physics"},
keywords = {Srni (Czechoslovakia); Proceedings; Winter school; Geometry; Physics},
location = {Palermo},
pages = {[135]-142},
publisher = {Circolo Matematico di Palermo},
title = {The relation between the dual and the adjoint Radon transforms},
url = {http://eudml.org/doc/220237},
year = {1991},
}

TY - CLSWK
AU - Cnops, J.
TI - The relation between the dual and the adjoint Radon transforms
T2 - Proceedings of the Winter School "Geometry and Physics"
PY - 1991
CY - Palermo
PB - Circolo Matematico di Palermo
SP - [135]
EP - 142
AB - [For the entire collection see Zbl 0742.00067.]Let $P^m$ be the set of hyperplanes $\sigma :\langle \vec{x},\vec{\theta }\rangle =p$ in $\mathcal {R}^m$, $S^{m-1}$ the unit sphere of $\mathcal {R}^m$, $E^m$ the exterior of the unit ball, $T^m$ the set of hyperplanes not passing through the unit ball, $Rf(\vec{\theta },p)=\int _\sigma f(\vec{x})d\vec{x}$ the Radon transform, $R^\#g(\vec{x})=\int _{S^{m-1}}g(\vec{\theta },\langle \vec{x},\vec{\theta }\rangle )dS_{\vec{\theta }}$ its dual. $R$ as operator from $L^2(\mathcal {R}^m)$ to $L^2(S^{m-1)}\times \mathcal {R})$ is a closable, densely defined operator, $R^*$ denotes the operator given by $(R^*g)(\vec{x})=R^\#g(\vec{x})$ if the integral exists for $\vec{x}\in \mathcal {R}^m$ a.e. Then the closure of $R^*$ is the adjoint of $R$. The author shows that the Radon transform and its dual can be linked by two operators of geometrical nature. Using the relation between the dual and the adjoint transform he obtains results regard!
KW - Srni (Czechoslovakia); Proceedings; Winter school; Geometry; Physics
UR - http://eudml.org/doc/220237
ER -