Almost integer translates. Do nice generators exist?

Alexander Olevskii*, Alexander Ulanovskii

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

26 Scopus citations

Abstract

It has been shown earlier by the first author that for any nonzero perturbation of the integers λn = n + 0(1), λn ≠ n, there is a generator, that is a function ψ ∈ L2(R) such that the system of translates {ψ(x - λn)} is complete in L2(R). We ask if ψ can be chosen with fast decay. We prove that in general it cannot. On the other hand, if the perturbations are 'quasianalytically small,' than it can, and this decay restriction is sharp. A certain class of complex measures which we call 'shrinkable' is introduced, and it is shown that the zeros sets of such measures do dot admit generators with fast decay.

Original languageEnglish
Pages (from-to)93-104
Number of pages12
JournalJournal of Fourier Analysis and Applications
Volume10
Issue number1
DOIs
StatePublished - 2004

Keywords

  • Beurling-Malliavin density
  • Exponential systems
  • Generators
  • Quasi-analyticity

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