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Title: The relation between the dual and the adjoint Radon transforms (English)
Author: Cnops, J.
Language: English
Journal: Proceedings of the Winter School "Geometry and Physics"
Issue: 1990
Pages: [135]-142
Category: math
Summary: [For the entire collection see Zbl 0742.00067.]\par Let $P\sp m$ be the set of hyperplanes $\sigma:\langle\vec x,\vec\theta\rangle=p$ in $\mathcal{R}\sp m$, $S\sp{m-1}$ the unit sphere of $\mathcal{R}\sp m$, $E\sp m$ the exterior of the unit ball, $T\sp m$ the set of hyperplanes not passing through the unit ball, $Rf(\vec\theta,p)=\int\sb \sigma f(\vec x)d\vec x$ the Radon transform, $R\sp \#g(\vec x)=\int\sb{S\sp{m-1}}g(\vec\theta,\langle\vec x,\vec\theta\rangle)dS\sb{\vec\theta}$ its dual. $R$ as operator from $L\sp 2(\mathcal{R}\sp m)$ to $L\sp 2(S\sp{m-1)}\times\mathcal{R})$ is a closable, densely defined operator, $R\sp*$ denotes the operator given by $(R\sp*g)(\vec x)=R\sp \#g(\vec x)$ if the integral exists for $\vec x\in\mathcal{R}\sp m$ a.e. Then the closure of $R\sp*$ 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! (English)
MSC: 44A12
MSC: 47B38
MSC: 53C65
idZBL: Zbl 0751.44001
idMR: MR1151898
Date available: 2009-07-13T21:26:52Z
Last updated: 2012-09-18
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