Lattice sub-tilings and frames in LCA groups

TL;DR

This paper characterizes when the set of characters associated with a dual lattice forms a frame for functions on a subset of a locally compact abelian group, linking it to the almost disjointness of translates of the subset.

Contribution

It provides a new characterization of frames generated by dual lattices in LCA groups, extending known results like Fuglede's theorem and offering a simple criterion for frames of modulates.

Findings

Frames form if and only if translates of the set are almost disjoint.

Reveals a connection between lattice duality and frame properties in LCA groups.

Provides a simple characterization for frames of modulates.

Abstract

Given a lattice $\Lambda$ in a locally compact abelian group $G$ and a measurable subset $\Omega$ with finite and positive measure, then the set of characters associated to the dual lattice form a frame for $L^2(\Omega)$ if and only if the distinct translates by $\Lambda$ of $\Omega$ have almost empty intersections. Some consequences of this results are the well-known Fuglede theorem for lattices, as well as a simple characterization for frames of modulates.