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

### Joaquim Bruna

Universitat Autonoma de Barcelona, Bellaterra, Spain### Alexander Olevskii

Tel Aviv University, Israel### Alexander Ulanovskii

Stavanger University College, Norway

## 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 $\varphi\in L^1(\mathbb R)$ such that its $\Lambda$-translates $\varphi(x-\lambda), \lambda\in\Lambda$, 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.

## Cite this article

Joaquim Bruna, Alexander Olevskii, Alexander Ulanovskii, Completeness in $L^1 (\mathbb R)$ of discrete translates. Rev. Mat. Iberoam. 22 (2006), no. 1, pp. 1–16

DOI 10.4171/RMI/447