Completeness in $L^1(R)$ of discrete translates

Abstract

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