Fourier Series pp 205-233 | Cite as

# Changing Signs of Fourier Coefficients

Chapter

## Abstract

In this chapter we shall be concerned with some remarkable facts concerning not one Fourier series
but rather “most” series
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
if this series converges, then “most” members of the family are, in particular, Fourier series of functions in

$$ \sum\limits_{{{\text{n}} \in {\text{Z}}}} {\hat{f}(n){e^{{{\text{inx}}}}}}, $$

$$ \sum\limits_{{{\text{n}} \in {\text{Z}}}} {\pm \hat{f}(n){e^{{{\text{inx}}}}}} $$

$$\sum\limits_{{\text{n}} \in {\text{Z}}} {|\hat f(n){|^{\text{2}}};}$$

**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

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## Copyright information

© Springer-Verlag, New York, Inc. 1982