On the generalized absolute convergence of Fourier series

Abstract

Sufficient conditions are given by means of the best trigonometric approxi- mation in $L^{p}(1<p\leq 2)$ and structural properties of $f\in L^{p}$ for the convergence of the series $$\sum_{n=1}^{\infty}\omega_{n}(\varphi(|a_{n}|)+\varphi(|b_{n}|)) ,$$ where $a_{n}$ and $b_{n}$ are the Fourier coefficients of $f$, $\{\omega_{n}\}$ is a certain sequence of positive numbers, $\varphi(u)(u\geq 0)$ denotes an increasing concave function.