Frames of exponentials and sub-multitiles in LCA groups

TL;DR

This paper explores conditions under which frames of exponential functions exist in L^2 spaces over LCA groups, linking sub-multitiling properties to the existence of such frames and extending previous results.

Contribution

It establishes a new connection between sub-multitiling properties and the existence of exponential frames in LCA groups, including converse results and extended conditions.

Findings

Sub-multitiling guarantees exponential frames in LCA groups.

Converse of the main result is proven, linking frames to tiling properties.

Conditions are extended from Riesz bases to general frames.

Abstract

In this note we investigate the existence of frames of exponentials for $L^2(\Omega)$ in the setting of LCA groups. Our main result shows that sub-multitiling properties of $\Omega \subset \widehat{G}$ with respect to a uniform lattice $\Gamma$ of $\widehat{G}$ guarantee the existence of a frame of exponentials with frequencies in a finite number of translates of the annihilator of $\Gamma$. We also prove the converse of this result and provide conditions for the existence of these frames. These conditions extend recent results on Riesz bases of exponentials and multitilings to frames.