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Fourier Series pp 234-276 | Cite as

Lacunary Fourier Series

  • R. E. Edwards
Part of the Graduate Texts in Mathematics book series (GTM, volume 85)

Abstract

As the name suggests, a lacunary trigonometric series is, roughly speaking, a trigonometric series \( \sum\nolimits_{{{\text{n\^I Z}}}} {{c_{{\text{n}}}}{e^{{{\text{inx}}}}}} \) in which c n = 0 for all integers n save perhaps those belonging to a relatively sparse subset E of Z. Examples of such series have appeared momentarily in Exercises 5.6 and 6.13. Indeed for the Cantor group ℒ, the good behaviour of a lacunary Walsh-Fourier series
$$ \sum\limits_{{\zeta\in\mathcal{R}}} {{c_{\zeta }}\zeta} $$
(whose coefficients vanish outside the subset ℛ of ℒ^) has already been noted: by Exercise 14.9, if the lacunary series belongs to C(ℒ) then it belongs to A(ℒ); and, by 14.2.1, if it belongs to L p (ℒ) for some p < 0, then it also belongs to L q (ℒ) for q ∈ [p, ∞]. In this chapter we shall be mainly concerned with lacunary Fourier series on the circle group and will deal more systematically with some (though by no means all) aspects of their curious behaviour.

Keywords

Arithmetic Progression Finite Union Infinite Subset Open Mapping Theorem Lacunary Series 
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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Copyright information

© Springer-Verlag, New York, Inc. 1982

Authors and Affiliations

  • R. E. Edwards
    • 1
  1. 1.Department of Mathematics (Institute for Advanced Studies)The Australian National UniversityCanberraAustralia

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