Frames generated by compact group actions

Abstract

Let K K be a compact group, and let ρ \rho be a representation of K K on a Hilbert space H ρ \mathcal {H}_\rho . We classify invariant subspaces of H ρ \mathcal {H}_\rho in terms of range functions, and investigate frames of the form { ρ ( ξ ) f i } ξ ∈ K , i ∈ I \{\rho (\xi ) f_i\}_{\xi \in K, i \in I} . This is done first in the setting of translation invariance, where K K is contained in a larger group G G and ρ \rho is left translation on H ρ = L 2 ( G ) \mathcal {H}_\rho = L^2(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.

Multiplication-invariant operators and the classification of LCA group frames

“…This enables us to apply the above mentioned results for MI operators and Bessel systems of multiplications to characterize abelian group frames. Similar results have been widely reported in the setting of translation-invariant spaces on abelian groups [8,10,12,14,15,40,43,51], as well as for representations of finite groups [56,57], discrete groups [7,55], and compact groups [41]. As a consequence, we obtain a classification (up to unitary equivalence) of abelian group frames with N ∈ N ∪ {∞} generators as positive, locally invertible, and integrable MI operators on L 2 ( Ĝ; ℓ 2 N ).…”

“…Theorem 2.4 generalizes to the case of countably many generators. For compact groups, a version of this theorem appeared as [41,Theorem 4.6].…”

The Zak transform and representations induced from characters of an abelian subgroup

“…Following the well-trodden path of [6], [7], [8], [9], the Zak transform serves to characterize right H-SI spaces in terms of range functions. Similar connections between the Zak transform and (left) shift-invariant spaces were first observed in [1], [10], and, independently, [11].…”

Spaces invariant under unitary representations of discrete groups

“…Other actions than translations were considered in [2,20], where the Zak transform is used to study the structure of spaces invariant under the action of an LCA group on a σ-finite measure space. The setting of compact groups was then treated in [21].…”