Uncertainty product for Vilenkin groups

Abstract

We study a localization of functions defined on Vilenkin groups. To measure the localization we introduce two uncertainty products $U{P}_{\lambda }$ and $U{P}_G$ that are similar to the Heisenberg uncertainty product. $U{P}_{\lambda }$ and $U{P}_G$ differ from each other by the metric used for the Vilenkin group $G$. We discuss analogs of a quantitative uncertainty principle. Representations for $U{P}_{\lambda }$ and $U{P}_G$ in terms of Walsh and Haar basis are given.

12 pages