Subspaces of L2(G) invariant under translation by an abelian subgroup

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 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].

Co-compact Gabor Systems on Locally Compact Abelian Groups

“…Theorem 3.1 can also be formulated for basic frames using the notion of range functions. A very general version of this result was obtained independently and concurrently in [28]. Theorem 3.1 is closely related to the theory of translation invariant subspaces which very recently has been studied in [4,28] using Zak transform methods (cf.…”

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

“…In [1], we extended the Zak transform construction of Weil [2] to analyze the left regular representation of a locally compact group, restricted down to an abelian subgroup. For the specific subgroup Z ≤ R, the construction of [1] reduces to what is usually called the Zak transform L 2 (R) → L 2 ([0, 1] 2 ), which features prominently in time-frequency analysis [3]. In the present setting, the Zak transform of [1] amounts to the operator Z ′ : L 2 (G) → L 2 (Ĥ; L 2 (G)) given by…”

“…For the specific subgroup Z ≤ R, the construction of [1] reduces to what is usually called the Zak transform L 2 (R) → L 2 ([0, 1] 2 ), which features prominently in time-frequency analysis [3]. In the present setting, the Zak transform of [1] amounts to the operator Z ′ : L 2 (G) → L 2 (Ĥ; L 2 (G)) given by…”

“…For each h ∈ H, let e h ∈ L ∞ (Ĥ) be the evaluation character given by e h (α) = α(h). In the language of[1], D := {e h } h∈H is a "Parseval determining set" for L 1 (Ĥ), by [1, Lemma 5.2]. Then [1, Theorem 2.10] gives the equivalence of (i') and (ii).In the case of a single generator A = {f }, Corollary 2 reduces to the following analogue of[13, Theorem 3].Corollary For any f ∈ L 2 (G) and any A, B > 0,{R h f } h∈His a frame for S({f }) with bounds A, B if and only if the following holds for every α ∈Ĥ: x∈Ω |(Z r f )(α)(x)| 2 ∈ {0} ∪ [A, B].…”

Modulation Preserving Operators on Locally Compact Abelian Groups

“…For a locally compact abelian (LCA) group G, a translation invariant space is defined to be a closed subspace of L 2 (G) that is invariant under translations by elements of a closed subgroup Γ of G. Translation invariant spaces in case of Γ closed, discrete and cocompact, called shift invariant spaces, have been studied in [4,5,12,13,14,17], and extended to the case of Γ closed and cocompact (but not necessarily discrete) in [3] (see also [8,9]). Recently, translation invariant spaces have been generalized in [2] to the case when Γ is closed (not necessarily discrete or cocompact), see also [11]. Another spaces, which are effective tools in Gabor theory, are spaces invariant under modulations.…”