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Analysis Mathematica

, Volume 44, Issue 2, pp 163–183 | Cite as

Convergence to Zero of Exponential Sums with Positive Integer Coefficients and Approximation by Sums of Shifts of a Single Function on the Line

  • P. A. Borodin
  • S. V. Konyagin
Article
  • 126 Downloads

Abstract

We prove that there is a sequence of trigonometric polynomials with positive integer coefficients, which converges to zero almost everywhere. We also prove that there is a function f: ℝ → ℝ such that the sums of its shifts are dense in all real spaces L p (ℝ) for 2 ≤ p < ∞ and also in the real space C0(R).

Key words and phrases

trigonometric polynomial with positive integer coefficients convergence almost everywhere approximation sum of shifts Lp space 

Mathematics Subject Classification

42A05 42A32 41A46 

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References

  1. [1]
    S. Banach, Théorie des opérations linéaires, Jacques Gabay (Sceaux, 1993).zbMATHGoogle Scholar
  2. [2]
    A. Bonami and Sz. Gy. Révész, Integral concentration of idempotent trigonometric polynomials with gaps, Amer. J. Math., 131 (2009), 1065–1108.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    P. A. Borodin, Density of a semigroup in a Banach space, Izv. Math., 78 (2014), 1079–1104.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    P. A. Borodin, Approximation by simple partial fractions with constraints on the poles. II, Sb. Math., 207 (2016), 331–341.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    P. A. Borodin, Approximation by sums of shifts of a single function on the circle, Izv. Math., 81 (2017), 1080–1094.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    P. Borwein and T. Erdélyi, Littlewood-type problems on subarcs of the unit circle, Indiana Univ. Math. J., 46 (1997), 1323–1346.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    M. Fekete, Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten, Math. Z., 17 (1923), 228–249.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    S. V. Konyagin, On a problem of Littlewood, Izv. Akad. Nauk SSSR Ser. Mat., 45 (1981), 243–265.MathSciNetzbMATHGoogle Scholar
  9. [9]
    S. V. Konyagin, On a question of Pichorides, C. R. Acad. Sci. Paris Ser. I Math., 324 (1997), 385–388.Google Scholar
  10. [10]
    O. C. McGehee, L. Pigno, and B. Smith, Hardy’s inequality and the L1-norm of exponential sums, Ann. of Math., 113 (1981), 613–618.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals, Clarendon Press (Oxford, 1937).zbMATHGoogle Scholar
  12. [12]
    J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces II, Springer (Berlin–Heidelberg–New York, 1979).CrossRefzbMATHGoogle Scholar
  13. [13]
    N. Wiener, Tauberian theorems, Ann. of Math. (2), 33 (1932), 1–100.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2018

Authors and Affiliations

  1. 1.Laboratory “Multivariate approximation and applications”, Department of Mechanics and MathematicsMoscow State UniversityMoscowRussia
  2. 2.Steklov Mathematical Institute of Russian Academy of SciencesMoscowRussia

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