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 $UP_{\lambda}$ and $UP_{G}$ that are similar to the Heisenberg uncertainty product. $UP_{\lambda}$ and $UP_{G}$ differ from each other by the metric used for the Vilenkin group $G$. We discuss analogs of a quantitative uncertainty principle. Representations for $UP_{\lambda}$ and $UP_{G}$ in terms of Walsh and Haar basis are given.