Counting function estimates for coherent frames and Riesz sequences

TL;DR

This paper derives asymptotic estimates for counting functions related to coherent frames and Riesz sequences, establishing density conditions and refining estimates under localization for groups of polynomial growth.

Contribution

It provides new asymptotic estimates for counting functions, extending density conditions to general unimodular amenable groups and improving precision with localization assumptions.

Findings

Recovered necessary density conditions for coherent frames and Riesz sequences.

Provided more precise estimates under localization conditions.

Extended results to groups of polynomial growth.

Abstract

We prove various estimates for the asymptotics of counting functions associated to point sets of coherent frames and Riesz sequences. The obtained results recover the necessary density conditions for coherent frames and Riesz sequences for general unimodular amenable groups, while providing more precise estimates under additional localization conditions on the coherent system for groups of polynomial growth.