Frames of exponentials and sub-multitiles in LCA groups

Abstract

In this note we investigate the existence of frames of exponentials for L 2 (Ω) in the setting of LCA groups. Our main result shows that sub-multitiling properties of Ω ⊂ G with respect to a uniform lattice Γ of G guarantee the existence of a frame of exponentials with frequencies in a finite number of translates of the annihilator of Γ. We also prove the converse of this result and provide conditions for the existence of these frames. These conditions extend recent results on Riesz bases of exponentials and multitilings to frames.

Band-limited functions, and Time-Frequency Analysis of Non-Stationary Signals: Application to Bioelectrical Signals Analysis

“…These spaces are also known as Paley-Wiener spaces in the literature. See [55], [8] for its connection with frames of translates and exponential bases. Note that in these references the exponentials are considered in the frequency domain, when dealing with systems of translates.…”

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. In this section we adapt this last result to the case where Ω is a finite measure set (not necessarily bounded) with the extra hypothesis of the admissibility.…”

“…The strategy of the proof in [4], consists in, given a bounded k-subtile Ω, enlarge it to obtain a k-tile ∆, and then select a structured Riesz basis of L 2 (∆), (that always exists in the bounded case for k-tiles). This basis, when restricted to Ω is a structured frame for L 2 (Ω).…”

“…In our case, since the k-subtile Ω is not necessarily bounded, we need to enlarge it to an admissible k-tile, to guarantee the existence of the Riesz basis. This requires an adaptation of the proof in [4]. This is done in the next proposition:…”