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.
Categories: math.CA