Changing Signs of Fourier Coefficients

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.