Completeness in $L^1(\mathbb{R})$ of discrete translates

Abstract

We characterize, in terms of the Beurling-Malliavin density, the discrete spectra $\Lambda \subset \mathbb{R}$ for which a generator exists, that is a function $\phi \in L^1(\mathbb{R})$ such that its $\Lambda$-translates $\phi (x-\lambda ),\lambda \in \Lambda$, span $L^1(\mathbb{R})$. It is shown that these spectra coincide with the uniqueness sets for certain analytic classes. We also present examples of discrete spectra $\Lambda \subset \mathbb{R}$ which do not admit a single generator while they admit a pair of generators.