JournalsrmiVol. 22, No. 1pp. 1–16

Completeness in L1(R)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
Completeness in $L^1 (\mathbb R)$ of discrete translates cover
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Abstract

We characterize, in terms of the Beurling-Malliavin density, the discrete spectra ΛR\Lambda\subset\mathbb R for which a generator exists, that is a function φL1(R)\varphi\in L^1(\mathbb R) such that its Λ\Lambda-translates φ(xλ),λΛ\varphi(x-\lambda), \lambda\in\Lambda, span L1(R)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 ΛR\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 L1(R)L^1 (\mathbb R) of discrete translates. Rev. Mat. Iberoam. 22 (2006), no. 1, pp. 1–16

DOI 10.4171/RMI/447