Lattice sub-tilings and frames in LCA groups

Abstract

Given a lattice Λ in a locally compact abelian group G and a measurable subset Ω with finite and positive measure, then the set of characters associated to the dual lattice form a frame for L 2 (Ω) if and only if the distinct translates by Λ of Ω have almost empty intersections. Some consequences of this results are the well-known Fuglede theorem for lattices, as well as a simple characterization for frames of modulates.

Frames of exponentials and sub-multitiles in LCA groups

“…Remark 2.6. Theorem 1.1 for the case ℓ = 1 can be found in [2]. In this case, the proof does not require making use of either the Paley-Wiener space of Ω or the range function associated to it as in the proof given above.…”

Three Problems on Exponential Bases

“…Next, we investigate conditions for which E(Z d ) is a frame on D. The following result is proved in [16,Lemma 2.10]. See also the recent [7,Theorem 2].…”

“…The connections between tiling and exponential bases are deep and interesting and have been intensely investigated. We refer the reader to the introduction and to the references cited in [7]. See also [22].…”

Riesz bases of exponentials on unbounded multi-tiles

“…The relation between k-subtiles and frames of exponentials was first studied in [5] for the case when Ω is a 1-subtile of finite measure in the context of locally compact abelian groups. Later on, it was proved in [4] that if Ω is a bounded k-subtile, then it admits an structured frame of exponentials.…”