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Analysis Mathematica

, Volume 44, Issue 2, pp 251–261 | Cite as

Discrete Translates in Function Spaces

  • A. Olevskii
  • A. Ulanovskii
Article

Abstract

For a wide class of separable Banach function spaces X on ℝ, we construct a Schwartz function φ such that the set of translates {φ(t − λ), λ ∈ Λ} spans X whenever Λ is an exponentially small perturbation of integers. Essentially, the only exception is the space L1(ℝ), which cannot be spanned by a uniformly discrete translates of a single function.

Key words and phrases

completeness of translates uniformly discrete set tempered distribution 

Mathematics Subject Classification

primary 42A65 secondary 46F99 

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References

  1. [1]
    A. Atzmon and A. Olevskii, Completeness of integer translates in function spaces on ℝ, J. Approx. Theory, 87 (1996), 291–327.MathSciNetCrossRefGoogle Scholar
  2. [2]
    A. Beurling, On a closure problem, Ark. Mat., 1 (1951), 301–303; see also: The Collected Works of Arne Beurling, vol. 2, Harmonic Analysis (Edited by L. Carleson, P. Malliavin, J. Neuberger and J. Wermer), Contemporary Mathematicians, Birkhuser (Boston, MA, 1989).MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    F. J. Friedlander and M. S. Joshi, Introduction to the Theory of Distributions, Cambridge University Press (1998).Google Scholar
  4. [4]
    L. Hörmander, The Analysis of Linear Partial Differential Operators, vol. I, Grundl. Math. Wissenschaft., 256, Springer (Berlin, Heidelberg, 1983).zbMATHGoogle Scholar
  5. [5]
    S. G. Kreĭn, Ju. I. Petunin and E. M. Semenov, Interpolation of Linear Operators, Transl. Math. Monographs, vol. 54, Amer. Math. Soc. (Providence, R.I., 1982).Google Scholar
  6. [6]
    N. Lev and A. Olevskii, Wiener’s “closure of translates” problem and Piatetski–Shapiro’s uniqueness phenomenon, Ann. of Math. (2), 174 (2011), 519–541.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    A. Olevskii, Completeness in L2(R) of almost integer translates, C. R. Acad. Sci. Paris, Sér. I Math., 324 (1997), 987–991.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    A. Olevskii and A. Ulanovskii, Almost integer translates. Do nice generators exist?, J. Fourier Anal. Appl., 10 (2004), 93–104.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    A. Olevskii and A. Ulanovskii, Functions with Disconnected Spectrum: Sampling, Interpolation, Translates, University Lecture Series, vol. 65, Amer. Math. Soc. (Providence, R.I., 2016).Google Scholar
  10. [10]
    A. Olevskii and A. Ulanovskii, Completeness of translates in Lp(R), Bull. London Math. Soc., DOI 10.1112/blms.12159.Google Scholar
  11. [11]
    J. Ramanathan and T. Steger, Incompleteness of sparse coherent states, Appl. Comput. Harmon. Anal., 2 (1995), 148–153.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    N. Wiener, Tauberian theorems, Ann. of Math., 33 (1932), 1–100.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2018

Authors and Affiliations

  1. 1.School of MathematicsTel Aviv UniversityRamat AvivIsrael
  2. 2.Stavanger UniversityStavangerNorway

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