Subspace dual and orthogonal frames by action of an abelian group

In this article, we discuss subspace duals of a frame of translates by an action of a closed abelian subgroup \(\Gamma \) of a locally compact group \({\mathscr {G}}.\) These subspace duals are not required to lie in the space generated by the frame. We characterise translation-generated subspace duals of a frame/Riesz basis involving the Zak transform for the pair \(({\mathscr {G}}, \Gamma ).\) We continue our discussion on the orthogonality of two translation-generated Bessel pairs using the Zak transform, which allows us to explore the dual of super-frames. As an example, we extend our findings to splines, Gabor systems, p-adic fields \({\mathbb {Q}} p,\) locally compact abelian groups using the fiberization map.