Completeness of discrete translates in the Hardy space 𝐻¹(ℝ)

Abstract

We provide a characterization of discrete sets Λ ⊂ R \Lambda \subset \mathbb {R} that admit a function whose Λ \Lambda -translates are complete in the Hardy space H 1 ( R ) H^1(\mathbb {R}) . In particular, we show that such a set cannot be uniformly discrete. We then give a uniformly discrete Λ ⊂ R \Lambda \subset \mathbb {R} which admits a pair of functions such that their Λ \Lambda -translates are complete in H 1 ( R ) H^1(\mathbb {R}) .