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Generalization on some theorems of L1-convergence of certain trigonometric series

It this paper we study L1 - convergence of the r-th derivatives of Fourier series with complex-valued coefficients. Namely new necessary - sufficient conditions for L1 - 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 $(BV)^\sigma_r, \sigma > 0, r=0, 1, 2, \cdots$ we generalize also the theorem proved by Garrett, Rees and Stanojević in[5]. Finally, for $\sigma = 1$ some corollaries of this theorem are given.

L1-convergence; trigonometric series; r-th derivatives of Fourier series; Wang-Telyakovskii class; Sheng class; Fomin class.

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