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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