Frame potential and finite abelian groups

TL;DR

This paper extends the characterization of convolutional tight frames from cyclic groups to general finite abelian groups, analyzing their structure via frame potential and subgroup translations.

Contribution

It generalizes previous results on tight frames from cyclic to all finite abelian groups, introducing new subgroup-based translation methods and norm conditions.

Findings

Tight frames with subgroup translations are characterized as local minimizers of the frame potential.

Analogues of downsampling and upsampling operators are developed for arbitrary finite abelian groups.

The study proposes directions for further research in frame theory and group structures.

Abstract

This article continues a prior investigation of the authors with the goal of extending characterization results of convolutional tight frames from the context of cyclic groups to general finite abelian groups. The collections studied are formed by translating a number of \emph{generators} by elements of a fixed subgroup and it is shown, under certain norm conditions, that tight frames with this structure are characterized as local minimizers of the frame potential. Natural analogs to the downsampling and upsampling operators of finite cyclic groups are studied for arbitrary subgroups of finite abelian groups. Directions of further study are also proposed.