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Functional Analysis and Its Applications

, Volume 52, Issue 1, pp 35–44 | Cite as

Summation of Unordered Arrays

  • E. V. Shchepin
Article
  • 20 Downloads

Abstract

An approach to the summation of unordered number and matrix arrays based on ordering them by absolute value (greedy summation) is proposed. Theorems on products of greedy sums are proved. A relationship between the theory of greedy summation and the theory of generalized Dirichlet series is revealed. The notion of asymptotic Dirichlet series is considered.

Key words

greedy sum unordered sum theorem on multiplications of sums generalized Dirichlet series asymptotic Dirichlet series Riesz means generic zeta-function 

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References

  1. [1]
    H. Bohr, “Über die Summabilität Dirichlet’scher Reihen,” Gött. Nachr., 1909, 247–262.Google Scholar
  2. [2]
    G. H. Hardy and M. Riesz, The general theory of Dirichlet’s series, Cambridge Tracts in Mathematics and Mathematical Physics, vol. 18, Cambridge University Press, Cambridge, 1915.Google Scholar
  3. [3]
    S. V. Konyagin and V. N. Temlyakov, “Convergence of greedy approximations I. General systems,” Studia Math., 159:1 (2003), 143–160.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    E. Landau, “Über dieMultiplikation Dirichlet’scher Reihen,” Rendiconti di Palermo, 24 (1907), 81–160.CrossRefGoogle Scholar
  5. [5]
    M. Riesz, “Sur la summation des séries de Dirichlet,” Comptes Rendus, 149 (1909), 18–21.zbMATHGoogle Scholar
  6. [6]
    T. J. Stiltijes, “Note sur la multiplication de deux séries,” Nouvelles Annales, ser. 3, 6 (1887), 210–215.Google Scholar
  7. [7]
    R. Graham, D. Knuth, and O. Patashnik, Concrete Mathematics: A Foundation of Computer Science, Addison-Wesley, Boston, 1994.zbMATHGoogle Scholar
  8. [8]
    A. F. Leont’ev, Series of Exponentials [in Russian], Nauka, Moscow, 1976.Google Scholar
  9. [9]
    P. L. Ul’yanov, “On the A-convergence of series,” Dokl. Ross. Akad. Nauk, 368:2 (1999), 160–163; English transl.: Russian Acad. Sci. Dokl. Math., 60:2 (1999), 176–179.MathSciNetzbMATHGoogle Scholar
  10. [10]
    P. L. Ul’yanov, “On properties of series,” Dokl. Ross. Akad. Nauk, 373:3 (2000), 307–310; English transl.: Russian Acad. Sci. Dokl. Math., 62:1 (2000), 56–59.MathSciNetzbMATHGoogle Scholar
  11. [11]
    E. Shchepin, Greedy sums and Dirichlet series, http://arxiv.org/abs/1110.5285.Google Scholar
  12. [12]
    E. Shchepin, Riesz means and greedy sums, http://arxiv.org/abs/1212.0821.Google Scholar
  13. [13]
    E. Steinitz, “Bedingt konvergente Reihen und konvexe Systeme,” J. Reine Angew. Math., 143 (1913), 128–175; ibid, 144 (1914), 1–40; ibid, 146 (1916), 1–52.MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Steklov Mathematical Institute of Russian Academy of SciencesMoscowRussia

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