The relation between the dual and the adjoint Radon transforms

Cnops, J.

Proceedings of the Winter School "Geometry and Physics", GDML_Books, (1991), p. [135]-142 / Harvested from

Résumé

[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!

Détail

MR MR1151898 | Zbl 0751.44001

EUDML-ID : urn:eudml:doc:220237
Mots clés: