Democratic systems of translates

Characterizations of democratic systems of translates on locally compact abelian groups

“…In this section we first describe the setting in which we formulate the questions mentioned in the introduction. Then we raise the main results from [11] to the level of abstract abelian groups.…”

“…When does F ψ constitute a democratic system in L 2 (G) in the sense of the general definition (1.1)? The first step towards the resolution is the following straightforward generalization of [11,Thm. 4.3].…”

“…Let us give a few words about the organization of the present paper. In Section 2 we generalize the results from [11] at the level of arbitrary locally compact abelian groups. Theorem 1 characterizes democratic systems of translates by testing sizes of the finite sums k∈Γ ψ(· − k) for all nonempty finite sets Γ ⊆ L. Theorem 3 characterizes democratic systems of translates in terms of boundedness and certain density properties of the generalization of the periodization function p ψ .…”

“…It is suspected that there is a lot of redundancy in verifying those conditions. However, no "operative" equivalent conditions are known, even in the case of integer translates, and only a conjecture was stated in the prequel to this paper [11]. Consequently, Section 3 discusses the possible operative conditions for torsion lattices, but mostly through a series of remarks, examples, counterexamples, and numerical data.…”

On quasi-greedy bases associated with unitary representations of countable groups

“…This idea was introduced in this context in [13]. For another application of this approach see [8,Theorem 4.7]. We first consider some general properties of quasi-greedy system that will be used later to obtain information about the function [ψ, ψ] associated with the cyclic subspace S given by (1.1).…”