Comments on the absolute convergence of Fourier series

Abstract

We give sufficient conditions for the convergence of the series having the following form $$\sum_{k=1}^{\infty}k^{\delta}(\varphi(|a_{n_{k}}|)+\varphi(|b_{n_{k}}|)) ,$$ where $a_{k}$ and $b_{k}$ are Fourier coefficients, $\delta\geq 0$, $\varphi(u)(u\geq 0)$ is an increasing concave function, and $\{n_{k}\}$ is a certain increasing sequence of natural numbers.