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1. CMB 2009 (vol 52 pp. 627)

Yu, Dan Sheng; Zhou, Ping; Zhou, Song Ping
 On $L^{1}$-Convergence of Fourier Series under the MVBV Condition Let $f\in L_{2\pi }$ be a real-valued even function with its Fourier series $% \frac{a_{0}}{2}+\sum_{n=1}^{\infty }a_{n}\cos nx,$ and let $S_{n}(f,x) ,\;n\geq 1,$ be the $n$-th partial sum of the Fourier series. It is well known that if the nonnegative sequence $\{a_{n}\}$ is decreasing and $\lim_{n\rightarrow \infty }a_{n}=0$, then% \begin{equation*} \lim_{n\rightarrow \infty }\Vert f-S_{n}(f)\Vert _{L}=0 \text{ if and only if }\lim_{n\rightarrow \infty }a_{n}\log n=0. \end{equation*}% We weaken the monotone condition in this classical result to the so-called mean value bounded variation (MVBV) condition. The generalization of the above classical result in real-valued function space is presented as a special case of the main result in this paper, which gives the $L^{1}$% -convergence of a function $f\in L_{2\pi }$ in complex space. We also give results on $L^{1}$-approximation of a function $f\in L_{2\pi }$ under the MVBV condition. Keywords:complex trigonometric series, $L^{1}$ convergence, monotonicity, mean value bounded variationCategories:42A25, 41A50
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