Frames of translates for model sets

Abstract

We study spanning properties of a family of functions translated along simple model sets. We characterize tight frame and dual frame generators for such irregular translates and we apply the results to Gabor systems. We use the connection between model sets and almost periodic functions and rely strongly on a Poisson summations formula for model sets to introduce the so-called bracket product, which then plays a crucial role in our approach. As a corollary to our main results we obtain a density statement for semi-regular Gabor frames.

Frames of Translates for Number-Theoretic Groups

“…for all g ∈ span{τ [x],C f ) : [x] ∈ G/H}. We then apply Lemma 4.2 directly to obtain (22) for all g ∈ V C,f , the desired conclusion.…”

“…One of these is the frame theoretic characterization of a closed span of translations. This is our bailiwick here, and it has become a topic with great generalization, applicability, intricacy, and abstraction, and with a large number of contributors, see, e.g., [8], [13], [17], [16], [12], [11], [1], [2], [22] and the references therein.…”