# System bandwidth and the existence of generalized shift-invariant frames

## TL;DR

This paper investigates the existence of tight frames in generalized shift-invariant systems over locally compact abelian groups, introducing the concept of system bandwidth and establishing conditions for frame existence based on analytical and algebraic properties.

## Contribution

It develops a new approach using unconditional convergence to study generalized shift-invariant frames, introduces system bandwidth as a key measure, and provides characterizations and conditions for frame generators.

## Key findings

- Unconditional convergence replaces local integrability in frame analysis.
- System bandwidth is introduced as a measure of a system's capacity.
- Counterexamples show orthonormal bases can have arbitrarily small bandwidth.

## Abstract

We consider the question whether, given a countable system of lattices $(\Gamma_j)_{j \in J}$ in a locally compact abelian group $G$, there exists a sequence of functions $(g_j)_{j \in J}$ such that the resulting generalized shift-invariant system $(g_j(\cdot - \gamma))_{j \in J, \gamma \in \Gamma_j}$ is a tight frame of $L^2(G)$. This paper develops a new approach to the study of almost periodic functions for generalized shift-invariant systems based on an \emph{unconditionally convergence property}, replacing previously used local integrability conditions. From this theory, we derive characterizing relations for tight and dual frame generators, we introduce the \emph{system bandwidth} as a measure of the total bandwidth a generalized shift-invariant system can carry, and we show that the so-called Calder\'on sum is uniformly bounded from below for generalized shift-invariant frames. We exhibit a condition on the lattice system for which the unconditionally convergence property is guaranteed to hold. Without the unconditionally convergence property, we show, counter intuitively, that even orthonormal bases can have arbitrary small system bandwidth. Our results show that the question of existence of frame generators for a general lattice system can be rather subtle, depending on analytical properties, such as the system bandwidth, as well as on algebraic properties of the lattice system.