Linear independence of translates implies linear independence of affine Parseval frames on LCA groups

TL;DR

This paper proves that, under certain conditions, linear independence of group translates of functions in L^2(G) guarantees the linear independence of affine Parseval frames on second countable, locally compact abelian groups.

Contribution

It establishes a link between the linear independence of translates and affine Parseval frames in the setting of LCA groups, extending wavelet frame results.

Findings

Linear independence of translates implies affine Parseval frame independence

Compatibility conditions between dilation and translation actions are crucial

Results extend wavelet frame theory to LCA groups

Abstract

Motivated by Bownik and Speegle's result on linear independence of wavelet Parseval frames, we consider affine systems (analogous to wavelet systems) defined on a second countable, locally compact abelian group $G$, where the translations are replaced by the action of a countable, discrete subgroup $\Gamma$ of $ G$ acting as a group of unitary operators on $L^2(G)$. The dilation operation in the wavelet setting is replaced by integer powers of a unitary operator $\delta$ onto $L^2(G)$. We show that, under some compatibility conditions between $\delta$ and the action of the group $\Gamma$, the linear independence of the translates of any function in $L^2(G)$ by elements of $\Gamma$ implies the linear independence of affine Parseval frames in $L^2(G)$.