On Sidon-Telyakovskii-type conditions for the integrability of multiple trigonometric series

Abstract

For the trigonometric series $$\sum_{k=0}^{\infty}a_k\sum_{l\in kV \setminus (k-1)V}e^{i(l, x)}, \quad a_k\rightarrow 0,\quad k\rightarrow \infty,$$ given on $[-\pi, \pi)^m$, where $V$ is some polyhedron in $R^m$, we prove that the inequality $$\int\limits_{T^m}\left|\sum^{\infty}_{k=0} a_k \sum_{l\in kV\setminus(k-1)V}e^{i(l, x)} \right| dx \leq C \sum^{\infty}_{k=0} (k+1) |\Delta A_k|,$$ holds if the coefficients $a_k$ satisfy the following conditions of the Sidon - Telyakovskii type: $$A_k\rightarrow\infty,\quad |\Delta a_k| \leq A_k, \quad \forall k \geq 0, \quad \sum^{\infty}_{k=0}(k+1) |\Delta A_k|<\infty.$$