Lacunary Fourier Series

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.