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 φgb)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 ast (g) = {g(. + γ)}γ∈γwe 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.