zbMATH — the first resource for mathematics

Translation invariant Radon transforms. (English) Zbl 0748.44003
Let \(R_ \mu\) be the generalized Radon transform and \(R^ t_ \mu\) the generalized dual Radon transform, i.e. \((R_ \mu f)(\omega,p)=\int_{\omega\centerdot x=p}f(x)\mu(x,\omega,p)dx\), \((R^ t_ \mu f)(x)=\int_{S^{n- 1}}f(\omega,x\centerdot\omega)\mu(x,\omega,\omega\centerdot x)d\omega\). These transforms are called exponential if \(\mu(x,\omega,p)=\mu_ 1(\omega,p)e^{\mu_ 2(\omega)\centerdot x}\) and translation invariant if \((f_ a(x)=f(a+x))\) \((R_ \mu f_ a)(\omega,p)=\nu(a,\omega,p)(R_ \mu f)(\omega,p+\omega\centerdot a)\), \((R^ t_ \mu f)_ a=R^ t_ \mu(f_{\omega\centerdot a}\nu(a,\centerdot,\centerdot))\) with a suitable function \(\nu\).
It is shown that \(R_ \mu(R^ t_ \mu)\) is exponential if and only if \(R_ \mu(R^ t_ \mu)\) is translation invariant. Conditions for \(R^ t_ \lambda\circ R_ \mu\) to be translation invariant (in the usual sense) are given. For exponential \(R_ \mu\) an inversion formula and a support theorem are proved.
Reviewer: F.Natterer
44A12 Radon transform