# Article

 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" Volume: Issue: 1990 Year: 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 Stable URL: http://hdl.handle.net/10338.dmlcz/701486 .

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