KASS-seminarium

Abstrakt finns enbart på engelska: Helgason's support theorem is of fundamental importance in integral geometry and has applications in tomography and image processing in general. It states that if a continuous function $u$ on ${\mathbb R}^n$ is rapidly decreasing at infinity and has vanishing integrals along every hyperplane not intersecting a given compact convex set $K$, then the support of $u$ is contained in $K$.  The hyperplane Radon transformation ${\mathcal R}u$ is a function defined on $\S^{n-1}\times \R$ such that ${\mathcal R}u(\omega,p)$ is the integral of $u$ over the hyperplane $\langle x,\omega\rangle=p$. 

In the lecture I will deal with the problem of locating the support of $u$ in case it is only known that for $\omega$ in a given subset $E$ of $\S^{n-1}$ the function $\R\ni p\mapsto {\mathcal R}u(\omega,p)$ has support in the compact interval $[a_\omega,b_\omega]$.  The main result is that if the set ${\mathbb C}E$, the union of all complex lines through the origin and points in $E$, is non-pluripolar, then the support of $u$ is located in a compact convex set which can be described in terms of the so-called homogeneous capacity of the set $E$ and the function $\sigma(\omega)=\max\{-a_\omega,b_\omega\}$ on $E$. Helgason's support theorem follows in the special case $E=\S^{n-1}$.