On uniformly minimal and 'uniformly complete' exponential systems

TL;DR

This paper extends density results for exponential systems, demonstrating their applicability to any positive measure set and introducing 'uniformly complete systems' as a relaxed frame concept.

Contribution

It generalizes previous results to broader sets and introduces the concept of 'uniformly complete systems' with analogous density properties.

Findings

Density results hold for any positive measure set.

Introduction of 'uniformly complete systems' as a relaxed frame.

Analogous density scales established for these systems.

Abstract

A.Olevskii and A.Ulanovskii obtained a scale of density results, which correspond to how well an exponential system approximates a uniformly minimal system over a compact set. We extend their result in several directions. First, we show that it holds for any set of positive finite measure. Next, we consider a relaxed version of frames, which we term 'uniformly complete systems', and obtain an analogues scale of density results for such systems.