On the Frames of Translates on Locally Compact Abelian Groups

For a second countable locally compact abelian group G, we study a system of translates generated by g ∈ L2(G). We find some equivalent conditions of this family to have some fundamental frame properties. More precisely, let Γ be a uniform lattice in G (a closed subgroup which is cocompact and discrete) and Γbe the annihilator of Γ in G. For g ∈ L2(G), the Γ-periodic function Φis defined as Φ(ξ) = ∑Γ∗|g(ξ+γ)|2 on Γ (the dual group of Γ) and some of its properties are investigated. In particular, it is shown that if Φis continuous, then the family {g(. + γ)}Γ can not be a redundant frame. Among other things it is shown that there is an isometry from L2(G) into L2(Γ) in such a way that the system of translates in L2(Γ) is transferred to a nice Fourier-like system in L2(Γ). Also, the canonical and oblique duals of the frames of translates are investigated.