This article discusses the existence conditions of dual frames and their uniqueness for the \(\Gamma \) -translation generated frames systems in \(L^2(\mathscr {G}),\) where \(\mathscr {G}\) be a locally compact group, and \(\Gamma \) is a closed abelian subgroup of \(\mathscr {G}.\) Necessary and sufficient conditions of such duals associated with the infimum cosine angles are obtained. The problems are first addressed in the context of multiplication-generated frame systems in \(L^2(X; {\mathcal {H}})\) , where X is a \(\sigma \) -finite measure space, then obtained the results for locally compact group setups.
