The relation between the dual and the adjoint Radon transforms. (English) Zbl 0751.44001

Geometry and physics, Proc. Winter Sch., Srni/Czech. 1990, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 26, 135-142 (1991).
[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 \(\mathbb{R}^ m\), \(S^{m-1}\) the unit sphere of \(\mathbb{R}^ m\), \(E^ m\) 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 \(L^ 2(\mathbb{R}^ m)\) to \(L^ 2(S^{m-1)}\times\mathbb{R})\) 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 \(\vec x\in\mathbb{R}^ m\) 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 regarding continuity, compactness and singular value decomposition in the cases of weighted \(L^ 2\)-spaces and \[ R:L^ 2(E^ m,(r^ 2-1)^{\alpha+(m-1)/2}r^{- 2\alpha})\to L^ 2(T^ m,(p^ 2-1)^ \alpha p^{1-m-2\alpha}), \]
\[ R:L^ 2(\mathbb{R}^ n,r^{m-1}e^{-1/2r^ 2})\to L^ 2(P^ m,p^{1- m}e^{-1/2p^ 2}). \]
Reviewer: W.Luther (Aachen)


44A12 Radon transform