Uncertainty product for Vilenkin groups

TL;DR

This paper investigates uncertainty products on Vilenkin groups, introducing two measures similar to the Heisenberg uncertainty principle, and explores their properties and representations using Walsh and Haar bases.

Contribution

It introduces two new uncertainty products for Vilenkin groups and analyzes their properties and representations, extending the concept of localization in harmonic analysis.

Findings

Defined two uncertainty products $UP_{\lambda}$ and $UP_{G}$ for Vilenkin groups.

Established analogs of the quantitative uncertainty principle for these products.

Provided representations of the products in terms of Walsh and Haar bases.

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.