Generalization on some theorems of $L^1$-convergence of certain trigonometric series

Abstract

In this paper we study $L^1$-convergence of the $r$-th derivatives of Fourier series with complex-valued coefficients. Namely new necessary-sufficient conditions for $L^1$-convergence of the $r$-th derivatives of Fourier series are given. These results generalize corresponding theorems proved by several authors (see [7], [10], [13], [19]). Applying the Wang-Telyakovskii class $({\bf B}{\bf V})_r^\sigma$, $\>\sigma>0$, $\>r=0,1,2,\ldots\,$ we generalize also the theorem proved by Garrett, Rees and Stanojevi\'{c} in [5]. Finally, for $\sigma=1$ some corollaries of this theorem are given.