On~lacunary series

I. M. Mikheev

Sbornik. Mathematics, Tome 27 (1975) no. 4, pp. 481-502

Résumé

This paper investigates properties of trigonometric $S_p$-systems ($p>2$) and Banach systems. In particular, the following theorems are established. Theorem 1. {\it Let the system $\{\cos n_kx,\sin n_kx\}$ be an $S_p$-system $(n_k$ integers, $p>2)$. Then if the series $a_0+\sum a_k\cos n_kx+b_k\sin n_k x$ converges on a set of positive measure it follows that $a_0^2+\sum a_k^2+b_k^2\infty$. If the same series converges to zero on a set of positive measure, all its coefficients are zero}. Theorem 2. {\it Let the system $\{\cos n_kx,\sin n_kx\}$ be a Banach system. Let $\alpha(\{n_k\},[a,b])$ be the number of terms of the sequence $\{n_k\}$ that lie on $[a,b]$. Then} $$ \lim_{h\to+\infty}\sup_a\frac{\alpha(\{n_k\},[a,a+h])}h=0. $$ Bibliography: 12 titles.