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

Joaquim Bruna·Alexander Olevskii·Alexander Ulanovskii
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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.

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Joaquim Bruna, Alexander Olevskii, Alexander Ulanovskii. 2003-07-24. Completeness in $L^1(R)$ of discrete translates. https://arxiv.org/abs/math/0307323

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