Completeness of Discrete Translates in $H^1(\mathbb{R})$

TL;DR

This paper characterizes which discrete sets allow for functions with complete translates in the Hardy space $H^1(R)$, revealing that such sets cannot be uniformly discrete but providing an example with a pair of functions.

Contribution

It offers a novel characterization of discrete sets enabling complete translates in $H^1(R)$ and constructs an explicit example with a pair of functions.

Findings

Sets admitting such functions are not uniformly discrete

A specific uniformly discrete set with a pair of functions is constructed

Complete translates can exist for certain discrete sets in $H^1(R)$

Abstract

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