Subspace Dual and orthogonal frames\\ by action of an abelian group

TL;DR

This paper explores subspace duals and orthogonality of frames generated by abelian group actions, extending classical results to various settings including splines, Gabor systems, and p-adic fields.

Contribution

It characterizes translation-generated subspace duals using the Zak transform and extends duality concepts to broader contexts like super-frames and different group structures.

Findings

Characterization of subspace duals via Zak transform.

Orthogonality conditions for translation-generated Bessel pairs.

Extension of results to splines, Gabor systems, and p-adic fields.

Abstract

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.