О коэффициентах рядо в Фурье по множествам

Abstract

The following statement is proved: Theorem.Let f(x), 0≦x≦2π, possess the Fourier expansion

$$\mathop \sum \limits_{\kappa = - \infty }^\infty c_\kappa e^{in} \kappa ^x with \bar c_\kappa = c_{ - \kappa } , n_\kappa = - \bar n_{ - \kappa }$$

where {n _k } is a Sidon sequence. Then in order to have

$$\mathop \sum \limits_{\kappa = - \infty }^\infty |c_\kappa |^p< \infty$$

for a given p, 1<p<2, it is necessary and sufficient that

$$\mathop \sum \limits_{k = 1}^\infty \left( {\frac{{\left\| f \right\|L^k (0,2\pi )}}{k}} \right)^p< \infty$$

.

An analogous statement holds true for series with respect to the Rademacher system.