Some Properties of Trigonometric Series Whose Terms have Random Signs

  • R. Salem
  • A. Zygmund
Part of the Mathematics and Its Applications book series (MAEE, volume 41)

Abstract

Trigonometric series of the type
$$ \sum\limits_1^{\infty } {{\varphi_n}(t)\left( {{a_n}\;\cos nx + {b_n}\;\sin nx} \right)} $$
(0.1)
where \( \left\{ {{\varphi_n}(t)} \right\} \) denotes the system of Rademacher functions, have been extensively studied in order to discover properties which belong to “almost all” series, that is to say which are true for almost all values of t.1 We propose here to add some new contributions to the theory.

Keywords

Trigonometric Series Absolute Constant Iterate Logarithm Random Sign Finite Range 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1a.
    Cf., in particular, Paley and Zygmund, Proc. Cambridge Phil. Soc., 26 (1930), pp. 337–357and 458–474CrossRefGoogle Scholar
  2. 1b.
    Cf., in particular, Paley and Zygmund, Proc. Cambridge Phil. Soc., 28 (1932), pp. 190–205.CrossRefGoogle Scholar
  3. 1.
    Cf. R. Salem, The absolute convergence of trigonometric series, Duke Math. Journal, 8 (1941), p. 333.Google Scholar
  4. 2.
    See Tamotsu Tsuchikura, Proc. of the Japan Academy, 27 (1951), pp. 141–145, and the results quoted there, especially Maruyama’s result.MathSciNetMATHCrossRefGoogle Scholar
  5. 1.
    See Bulletin des Sciences Mathématiques, 74 (1950).Google Scholar
  6. 1a.
    See Paley and Zygmund, loc. cit.CrossRefGoogle Scholar
  7. 1b.
    Zygmund, Trigonometrical Series, p. 125.Google Scholar
  8. 1.
    See also the authors’ notes “On lacunary trigonometric series” part I, Proc. Nat. Acad., 33 (1947), pp. 333–338, esp. p. 337, and part II, Ibid. 34 (1948), pp. 54–62.Google Scholar
  9. 1a.
    See Paley and Zygmund, loc. cit.CrossRefGoogle Scholar
  10. 1b.
    R. Salem, Comptes Rendus, 197 (1933), pp. 113–115Google Scholar
  11. 1c.
    R. Salem, Essais sur les séries trigonométriques, Paris (Hermann), 1940.Google Scholar

Copyright information

© Kluwer Academic Publishers 1989

Authors and Affiliations

  • R. Salem
  • A. Zygmund

There are no affiliations available

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