Revista Matemática Iberoamericana

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Volume 22, Issue 1, 2006, pp. 1–16
DOI: 10.4171/RMI/447

Published online: 2006-04-30

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

Joaquim Bruna[1], Alexander Olevskii[2] and Alexander Ulanovskii[3]

(1) Universitat Autonoma de Barcelona, Bellaterra, Spain
(2) Tel Aviv University, Israel
(3) Stavanger University College, Norway

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.

Keywords: Discrete translates, generator, Beurling-Malliavin density, uniqueness sets, Bernstein classes

Bruna Joaquim, Olevskii Alexander, Ulanovskii Alexander: Completeness in $L^1 (\mathbb R)$ of discrete translates. Rev. Mat. Iberoam. 22 (2006), 1-16. doi: 10.4171/RMI/447