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 realvalued 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 fS_{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 socalled
mean value bounded variation (MVBV) condition. The generalization of the
above classical result in realvalued 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 variation Categories:42A25, 41A50 
