Remarks on the uniform and \(L^1\)-convergence of trigonometric series
\textit{T. W. Chaundry and A. E. Jollife} [Lond. M. S. Proc. (2) 15, 214--216 (1916; JFM 46.0455.03)] proved that if \(a_k\) is a nonnegative sequence tending monotonically to zero, then the series \(\sum_{k=1}^\infty a_k \sin kx\) converges uniformly if and only if \(ka_k\to 0\) as \( k\to\infty\). This was extended by \textit{S. P. Zhou, P. Zhou and D. S. Yu} [Sci. China, Math. 53, No. ~7, 1853--1862 (2010; Zbl 1211.42006)] to more general classes of sequences. The author extends these results to even more general classes of sequences. He also extends a recent result of \textit{S. Tikhonov} [J. Math. Anal. Appl. 347, No. ~2, 416--427 (2008; Zbl 1257.42009)] on the \(L^1-\)convergence of Fourier series.
scientific article; zbMATH DE number 5857777