# Orthogonality and duality of frames over locally compact abelian groups

## TL;DR

This paper extends the theory of frames and duality for generalized translation invariant systems to super-spaces over locally compact abelian groups, providing new characterizations and applications to wave-packet systems.

## Contribution

It introduces super-GTI systems in super-spaces over LCA groups and characterizes dual frames, extending existing results to more general settings including wavelet and Gabor systems.

## Key findings

- Characterization of dual super-GTI frames in super-spaces
- Necessary and sufficient conditions for GTI-orthogonal frame systems
- Application to dual super wave-packet systems on LCA groups

## Abstract

Motivated by the recent work of Bownik and Ross \cite{BR}, and Jakobsen and Lemvig \cite{JL}, this article generalizes latest results on reproducing formulas for generalized translation invariant (GTI) systems to the setting of super-spaces over a second countable locally compact abelian (LCA) group $G$. To do so, we introduce the notion of a super-GTI system with finite sequences as generators from a super-space $L^2(G) \oplus \cdots \oplus L^2(G) $ ($N$ summands). We characterize the generators of two super-GTI systems in the super-space such that they form a super-dual frame pair. For this, we first give necessary and sufficient conditions for two Bessel families to be orthogonal frames (we call as GTI-orthogonal frame systems) when the Bessel families have the form of GTI systems in $L^2(G)$. As a consequence, we deduce similar results for several function systems including the case of TI systems, and GTI systems on compact abelian groups. As an application, we apply our duality result for super-GTI systems to the Bessel families with a wave-packet structure (combination of wavelet as well as Gabor structure), and hence a characterization for dual super wave-packet systems on LCA groups is obtained. In addition, we relate the well established theory from literature with our results by observing several deductions in context of wavelet and Gabor systems over LCA groups with $G=\mathbb{R}^d,\mathbb{Z}^d$, etc.