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Fourier Series pp 205-233 | Cite as

Changing Signs of Fourier Coefficients

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

Abstract

In this chapter we shall be concerned with some remarkable facts concerning not one Fourier series
$$ \sum\limits_{{{\text{n}} \in {\text{Z}}}} {\hat{f}(n){e^{{{\text{inx}}}}}}, $$
but rather “most” series
$$ \sum\limits_{{{\text{n}} \in {\text{Z}}}} {\pm \hat{f}(n){e^{{{\text{inx}}}}}} $$
of the family obtained by making random changes of sign in the coefficients of the original series. It turns out that the behaviour of “most” members of such a family depends solely on the convergence or divergence of the series
$$\sum\limits_{{\text{n}} \in {\text{Z}}} {|\hat f(n){|^{\text{2}}};}$$
if this series converges, then “most” members of the family are, in particular, Fourier series of functions in L p for every p < ∞; while, if this series diverges, “most” members of the family fail to be Fourier-Lebesgue (or even Fourier-Stieltjes) series at all. We shall concentrate principally on the good behaviour resulting from the assumed convergence of \( \sum {|\hat{f}(n){|^{2}};} \) results pertaining to the case in which \(\sum {|\hat f(n){|^2} = \infty }\) are mentioned only briefly in 14.2.3 and 14.3.5.

Keywords

Fourier Series FOURIER Coefficient Orlicz Space Compact Abelian Group Approximate Identity 
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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