$$\ell ^2(G)$$ ℓ 2 ( G ) -linear independence for systems generated by dual integrable representations of LCA groups

Abstract

Let T be a dual integrable representation of a countable discrete LCA group G acting on a Hilbert space \(\mathbb H\). We consider the problem of characterizing \(\ell ^2(G)\)-linear independence of the system \(\mathcal B_{\psi }=\{T_{g}\psi :g\in G\}\) for a given function \(\psi \in \mathbb H\) in terms of the bracket function. The characterization theorem is obtained for the case when G is a uniform lattice of the p-adic Vilenkin group acting by translations and a partial answer is given for the case when \(\mathcal B_{\psi }\) is the Gabor system.