Fourier Like Systems, Frame of Translates and their Oblique Duals on LCA-groups

The theory of frames of translates has an essential role in many areas of mathematics 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 L 2 (R) that has the form T (g) = {g(. − ak)}k∈Z where g ∈ L^2 (R) and a 0 are fixed. In the setting of L^2 (R), it is known that frames of translates can be characterized in terms of a 1-periodic function ([3, 6]). More precisely, for g ∈ L^2 (R), if we define Φg(ω) = P k∈Z |ϕb(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) ∞ A 0 , A =B , 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∗ in Gb (the dual group of G )[5, 7, 8, 10, 14−16]. For g ∈ L 2 (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 on Γ and investigate a characterization of trans- b lates of g ∈ L 2 (G) to have some properties. We achieve our goal by using an isometry from L 2 (G) into L 2 (Γ), in such a way that the system of translates in b L 2 (G) is transferred to a nice Fourier-like system in L 2 (Γ). To do so, we consider a fix b φ ∈ L 2 (Γ) b and define the Fourier-like system generated by φ as E(φ) = {Xγφ}γ∈Γ , where Xγ is the corresponding character γ on Γ. We deduce the structure of the canonical dual b 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.