Equality cases for the uncertainty principle in finite Abelian groups

TL;DR

This paper characterizes functions with minimal spectral support in certain finite Abelian groups, extending known equality cases of the uncertainty principle beyond previously studied groups and support sizes.

Contribution

It provides new characterizations of equality cases for the uncertainty principle in specific finite Abelian groups, including cases not previously understood.

Findings

Characterization for $ ext{support size}=k$ in $ ext{groups } ext{Z}/p ext{Z} imes ext{Z}/p ext{Z}$ and $ ext{Z}/p^2 ext{Z}$

Extension of equality cases to many values of $k$ in $ ext{Z}/p ext{Z} imes ext{Z}/q ext{Z}$

New insights into the structure of functions achieving minimal spectral support

Abstract

We consider the families of finite Abelian groups $\ZZ/p\ZZ\times \ZZ/p\ZZ$, $\ZZ/p^2\ZZ$ and $\ZZ/p\ZZ\times \ZZ/q\ZZ$ for $p,q$ two distinct prime numbers. For the two first families we give a simple characterization of all functions whose support has cardinality $k$ while the size of the spectrum satisfies a minimality condition. We do it for a large number of values of $k$ in the third case. Such equality cases were previously known when $k$ divides the cardinality of the group, or for groups $\ZZ/p\ZZ$.