On $L^{1}$-Convergence of Fourier Series Under $MVBV$ Condition

Let $fin L_{2pi}$ be a real-valued even function with its Fourier series $ frac{a_{0}}{2}+sum_{n=1}^{infty}a_{n}cos nx,$ and let $S_{n}(f,x), ngeq 1,$ be the $n$-th partial sum of the Fourier series. It is well-known that if the nonnegative sequence ${a_{n}}$ is decreasing and $limlimits_{nto infty}a_{n}=0$, then $$ limlimits_{nto infty}Vert f-S_{n}(f)Vert_{L}=0 {if and only if} limlimits_{nto infty}a_{n}log n=0. $$ 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 $fin L_{2pi}$ in complex space. We also give results on $L^{1}$-approximation of a function $fin L_{2pi}$ under the $% MVBV$ condition.