On Wiegerinck's support theorem

Let continuous function $f(x)$, $x\in\mathbb R^n$, tend to $0$ as $\|x\|\to\infty$ faster than any negative degree of $\|x\|$. Let Radon transform $\tilde f(\omega,t)$, $\omega\in\mathbb R^n$, $\|\omega\|=1$, $t\in\mathbb R$, of $f$ also tend to $0$ as $t\to\infty$ and, besides, do it very fast on a massive enough set of $\omega$. In the paper, we describe the additional properties that $f$ has under these assumptions for different rates of fast decreasing. In particular, the extremal case where $\tilde f(\omega,t)$ has the compact support with respect to $t$ for the open subset of unit sphere corresponds to Wiegerinck's Theorem mentioned in the title.