On $L^{1}$-Convergence of Fourier Series Under $MVBV$ Condition
| dc.creator | Yu, Dan Sheng | |
| dc.creator | Zhou, Ping | |
| dc.creator | Zhou, Song Ping | |
| dc.date | 2007-04-14 | |
| dc.date.accessioned | 2026-07-07T07:56:39Z | |
| dc.date.available | 2026-07-07T07:56:39Z | |
| dc.description | Let $f\in L_{2π}$ 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 $f\in L_{2π}$ in complex space. We also give results on $L^{1}$-approximation of a function $f\in L_{2π}$ under the $% MVBV$ condition. | |
| dc.description | 13 Pages, Accepted by Canad. Math. Bull | |
| dc.identifier | https://arxiv.org/abs/0704.1865 | |
| dc.identifier | http://arxiv.org/abs/0704.1865 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/127390 | |
| dc.subject | Classical Analysis and ODEs | |
| dc.subject | 42A25;41A50 | |
| dc.title | On $L^{1}$-Convergence of Fourier Series Under $MVBV$ Condition | |
| dc.type | text |