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

Abstract

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 $S_n(f,x),n\ge 1,$ be the $n$-th partial sum of the Fourier series. It is well-known that if the nonnegative sequence $\{a_n\}$ is decreasing and $\underset{n\to \infty }{\lim }{a}_n=0$, then $$\underset{n\to \infty }{\lim }\Vert f-S_n(f){\Vert }_L=0ifandonlyif\underset{n\to \infty }{\lim }{a}_n\log n=0.$$ 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 condition.

13 Pages, Accepted by Canad. Math. Bull