Subspaces of $L^2(G)$ invariant under translation by an abelian subgroup

TL;DR

This paper classifies subspaces of $L^2(G)$ invariant under translation by an abelian subgroup using a range function approach, introducing new transforms and tools for both abelian and non-abelian groups.

Contribution

It provides a novel classification framework for translation-invariant subspaces in $L^2(G)$, extending to non-abelian groups with new transforms and measure-theoretic tools.

Findings

Range function classification for invariant subspaces

Introduction of a Zak-like transform for non-abelian groups

Conditions for translation families to form frames or Riesz sequences

Abstract

For a second countable locally compact group $G$ and a closed abelian subgroup $H$, we give a range function classification of closed subspaces in $L^2(G)$ invariant under left translation by $H$. For a family $\mathscr{A} \subset L^2(G)$, this classification ties with a set of conditions under which the translations of $\mathscr{A}$ by $H$ form a continuous frame or a Riesz sequence. When $G$ is abelian, our work relies on a fiberization map; for the more general case, we introduce an analogue of the Zak transform. Both transformations intertwine translation with modulation, and both rely on a new group-theoretic tool: for a closed subgroup $\Gamma \subset G$, we produce a measure on the space $\Gamma \backslash G$ of right cosets that gives a measure space isomorphism $G \cong \Gamma \times \Gamma \backslash G$. Outside of the group setting, we consider a more general problem: for a measure space $X$ and a Hilbert space $\mathcal{H}$, we investigate conditions under which a family of functions in $L^2(X;\mathcal{H})$ multiplies with a basis-like system in $L^2(X)$ to produce a continuous frame or a Riesz sequence in $L^2(X;\mathcal{H})$. Finally, we explore connections with dual integrable representations of LCA groups, as introduced by Hern{\'a}ndez et al.