Zeros of the Zak transform on locally compact abelian groups

Abstract

Let G be a locally compact abelian group. The notion of Zak transform on L 2 (R d) extends to L 2 (G). Suppose that G is compactly generated and its connected component of the identity is non-compact. Generalizing a classical result for L 2 (R), we then prove that if f ∈ L 2 (G) is such that its Zak transform Zf is continuous on G × G, then Zf has a zero.

Riesz bases of exponentials and the Bohr topology

“…A Borel section of Γ/Λ is a set of representatives of this quotient that is a Borel set. It can be proved that there always exists a relatively compact Borel section P ⊆ Γ, which we will call a fundamental domain; see [4,16]. As Λ is countable and the Haar measure of Γ, denoted m, is Borel regular, we always have 0 < m(P ) < ∞.…”

A characterization of the Riesz family of shifts of functions on locally compact Abelian groups

“…Also in [15], it was seen that L 2 (G/L) ∼ = L 2 (S L ), when G is a second countable and LCA group and L is a uniform lattice in G.…”

“…For a uniform lattice L in G, a fundamental domain is a measurable set S L in G, such that every x ∈ G can be uniquely written as x = ks, for k ∈ L and s ∈ S L . The existence of a relatively compact fundamental domain for a uniform lattice in an LCA group G is guaranteed by [15,Lemma 2], and it has been shown that S L has positive measure (see [14], [15]). For a uniform lattice L, a closed subspace V ⊆ L 2 (G) is called L-invariant if it is invariant under translations by elements of L. In other…”

Deformations of Gabor frames on the adeles and other locally compact abelian groups

“…For instance, the integers Z, finite cyclic groups Z/nZ and the circle group T are important when sampling or periodizing Gabor frames on R [26,29,30,39]. Other considerations on general locally compact abelian groups include [21,26,32], and Balian-Low type phenomena were proved to hold for finite groups in [33].…”