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

Canadian mathematical bulletin

关键词: complex trigonometric series;    $L^{1}$ convergence;    monotonicity;    mean value bounded variation;   

DOI  :  10.4153/CMB-2009-061-6

学科分类：数学（综合）

来源: University of Toronto Press * Journals Division

【 摘 要 】

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. Itis well known that if the nonnegative sequence ${a_{n}}$ is decreasing and$lim_{nightarrow infty }a_{n}=0$, then%egin{equation*}lim_{nightarrow infty }Vert f-S_{n}(f)Vert _{L}=0ext{ ifand only if }lim_{nightarrow infty }a_{n}log n=0.end{equation*}%We weaken the monotone condition in this classical result to the so-calledmean value bounded variation (MVBV) condition. The generalization of theabove classical result in real-valued function space is presented as aspecial 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 giveresults on $L^{1}$-approximation of a function $fin L_{2pi }$ under theMVBV condition.

【 授权许可】

Unknown