On Wiegerinck's support theorem

Анотація

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, $ $t\in \mathbb{R}^n$ 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.