arXiv 0704.18652007-04-14math.CA

在$MVBV$条件下傅里叶级数的$L^{1}$ -收敛性

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

Dan Sheng Yu, Ping Zhou, Song Ping Zhou

备注：: 13 Pages, Accepted by Canad. Math. Bull

摘要：

设$f\in L_{2π}$是一个实值偶函数，它的傅里叶级数是$ \frac{a_{0}}{2}+\sum_{n=1}^{\infty}a_{n}\cos nx,$，而$S_{n}(f,x), n\geq 1,$是傅里叶级数的第$n$次部分和。众所周知，如果非负序列$\{a_{n}\}$是递减的且$\lim\limits_{n\to \infty}a_{n}=0$是递减的，则$$ \lim\limits_{n\to \infty}\Vert f-S_{n}(f)\Vert_{L}=0 {if and only if} \lim\limits_{n\to \infty}a_{n}\log n=0. $$我们将这个经典结果中的单调条件弱化为所谓的均值有界变差（$MVBV$）条件.本文给出了复空间中函数$f\in L_{2π}$的$L^{1}$ % -收敛性的主要结果，并将其推广到实值函数空间中作为其特例.在$% MVBV$条件下，给出了函数$f\in L_{2π}$的$L^{1}$ -逼近的结果.

英文摘要：

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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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