  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 $\lim\limits_{n\to
\infty}a_{n}=0$, then $$ \lim\limits_{n\to \infty}\Vert f-S_{n}(f)\Vert_{L}=0
{if and only if} \lim\limits_{n\to \infty}a_{n}\log n=0. $$ 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.
