Generalizations of the Sidon-Telyakovskiĭ theorem

Abstract:

The well-known Sidon-Telyakovskii integrability condition is considerably lightened as follows: \[ \frac {1}{n}\sum \limits _{k = 1}^n {\frac {{|\Delta c(k){|^p}}}{{A_k^p}} = O(1),\quad n \to \infty } ,\] where $\{ c(n)\}$ is a certain null-sequence and $1 < p \leq 2$. It is proved that $\sum \nolimits _{n = 1}^\infty {{n^{p - 1}}|\Delta c(n){|^p}{\rho ^p}(n) < \infty }$ is also a sufficient integrability condition provided $\sum \nolimits _{n = 1}^\infty {(1/n\rho (n)) < \infty }$, where $\{ \rho (n)\}$ is an increasing sequence of positive numbers.