Search

Having potential applications in multiplexing techniques and in the synthesis of frames, orthogonality (or strongly disjointness) plays a significant role in frame theory (e.g. construction of new frames from existing ones, constructions related with duality, etc.). In this article, we study orthogonality of a pair of frames over locally compact abelian (LCA) groups. We start with the investigation of the dual Gramian analysis tools of Ron and Shen through a pre-Gramian operator over the set-up of LCA groups. Then we fiberize some operators associated with Bessel families generated by unitary actions of co-compact (not necessarily discrete) subgroups of LCA groups. Using this fiberization, we study and characterize a pair of orthogonal frames generated by the action of a unitary representation ρ of a co-compact subgroup Γ ⊂ G on a separable Hilbert space L 2 (G) , where G is a second countable LCA group. Precisely, we consider frames of the form { ρ (γ) ψ : γ ∈ Γ , ψ ∈ Ψ } for a countable family Ψ in L 2 (G). We pay special attention to this problem in the context of translation-invariant space by assuming ρ as the action of Γ on L 2 (G) by left-translation. The representation of Γ acting on L 2 (G) by (left-)translation is called the (left-)regular representation of Γ. Further, we apply our results on co-compact Gabor systems over LCA groups. At this juncture, it is pertinent to note that the resulting characterization can be useful for constructing new frames by using various techniques including the unitary extension principle by Ron and Shen [24] and its recent extension to LCA groups by Christensen and Goh [7]. [ABSTRACT FROM AUTHOR] more...

Let K be a compact group, and let ρ be a representation of K on a Hilbert space 𝓗ρ. We classify invariant subspaces of 𝓗ρ in terms of range functions, and investigate frames of the form {ρ(ξ)fi}ξ∈K,i∈I. This is done first in the setting of translation invariance, where K is contained in a larger group G and ρ is left translation on 𝓗ρ = L2(G). For this case, our analysis relies on a new, operator-valued version of the Zak transform. For more general representations, we develop a calculational system known as a bracket to analyze representation structures and frames with a single generator. Several applications are explored. Then we turn our attention to frames with multiple generators, giving a duality theorem that encapsulates much of the existing research on frames generated by finite groups, as well as classical duality of frames and Riesz sequences. [ABSTRACT FROM AUTHOR] more...

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 A ⊆ L 2 ( G ) , this classification ties with a set of conditions under which the translations of 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 Γ ⊆ G , we produce a measure on the space Γ \ G of right cosets that gives a measure space isomorphism G ≅ Γ × Γ \ G . Outside of the group setting, we consider a more general problem: for a measure space X and a Hilbert space H , we investigate conditions under which a family of functions in L 2 ( X ; H ) multiplies with a basis-like system in L 2 ( X ) to produce a continuous frame or a Riesz sequence in L 2 ( X ; H ) . Finally, we explore connections with dual integrable representations of LCA groups, as introduced by Hernández et al. in [25] . [ABSTRACT FROM AUTHOR] more...

Let $$\Gamma $$ be a closed co-compact subgroup of a second countable locally compact abelian (LCA) group $$G$$ . In this paper we study translation-invariant (TI) subspaces of $$L^2(G)$$ by elements of $$\Gamma $$ . We characterize such spaces in terms of range functions extending the results from the Euclidean and LCA setting. The main innovation of this paper, which contrasts with earlier works, is that we do not require that $$\Gamma $$ be discrete. As a consequence, our characterization of TI-spaces is new even in the classical setting of $$G=\mathbb {R}^n$$ . We also extend the notion of the spectral function in $$\mathbb {R}^n$$ to the LCA setting. It is shown that spectral functions, initially defined in terms of $$\Gamma $$ , do not depend on $$\Gamma $$ . Several properties equivalent to the definition of spectral functions are given. In particular, we show that the spectral function scales nicely under the action of epimorphisms of $$G$$ with compact kernel. Finally, we show that for a large class of LCA groups, the spectral function is given as a pointwise limit. [ABSTRACT FROM AUTHOR] more...