Abstract The theory of frames of translates has an essential role in many areas of mathe- matics and its applications such as wavelet theory and reconstruction of signals from sample values [1 − 4, 6, 11, 12, 13]. A lattice system of translates is a sequence in L2(R) that has the form T (g) = {g(. − ak)}k∈Z where g ∈ L2(R) and a > 0 are fixed. In the setting of L2(R), it is known that frames of translates can be characterized in terms of a 1-periodic function ([3, 6]). More precisely, for g ∈ L2(R), if we define Φg (ω) = ∑ k∈Z |̂ ϕ(ω +k)|2, then Φg is a 1-periodic function which characterizes frames of translates as follows. (a) T (g) is a frame sequence if and only if there exist 0 < A ≤ B < ∞ such that A ≤ Φg ≤ B, a.e. on the zero set of Φg . (b) T (g) is a Riesz basis for the closure span of T (g) if and only if there exist 0 < A ≤ B < ∞ such that A ≤ Φg ≤ B, a.e. (c) T (g) is an orthonormal basis for the closure span of T (g) if and only if Φg = 1 a.e. Our goal in this presentation is a generalization of frames of translates in the setting of locally compact abelian groups. Let G be a locally compact abelian (LCA) group and Γ be a uniform lattice in G (i.e. a discrete subgroup of G which is co-compact), with the annihilator Γ∗ in̂ G (the dual group of G )[5, 7, 8, 10, 14 − 16]. For g ∈ L2(G), a system of translates generated by g via Γ, is defined as T (g) = {g(. + γ)}γ∈Γ We define a Γ∗-periodic function Φg on̂ Γ and investigate a characterization of trans- lates of g ∈ L2(G) to have some properties. We achieve our goal by using an isometry from L2(G) into L2(̂ Γ), in such a way that the system of translates in L2(G) is trans- ferred to a nice Fourier-like system in L2(̂ Γ). To do so, we consider a fix φ ∈ L2(̂ Γ) and define the Fourier-like system generated by φ as E(φ) = {Xγ φ}γ∈Γ, where Xγ is the corresponding character γ on̂ Γ. We deduce the structure of the canonical dual frame of a frame sequence T (g). Using the fact that the frame operator of a frame of translates commutes with the translation operator, it is shown that the canonical dual frame of T (g) has the same form T (h) for some h ∈ span(T (g)). Some properties of Φg which are useful in the study of the translates sequence generated by g are investigated. In particular, it is shown that if Φg is continuous, then T (g) can not be a redundant frame.

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