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

Convolution 

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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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