傅里叶级数的几乎处处可和性及其收敛集
Almost everywhere summability of Fourier series with indicating the set of convergence
摘要：
本文研究了下列问题：在函数$\int_0^x f$的导数存在的点上，哪些乘子$\{λ_{k, n}\}$保证函数$f\in L_1[-π, π]$ $$ \sum_{k=-\infty}^\infty λ_{k, n}\hat{f}_k e^{ikx}, $$的傅里叶级数的线性平均值作为$n\to \infty$的收敛，其中$\widehat{f}_k$是$k$阶傅里叶系数.的$(C, 1)$ -手段（$λ_{k, n}=(1-\frac {|k|}{n+1})_+$）的收敛性的标准被发现，而在一般情况下$λ_{k, n}=ϕ(\frac {k}{n+1})$的一个充分条件推导出在所有这样的点（即，几乎处处）的收敛性。答案是在属于$ϕ(x)$和$xϕ'(x)$绝对收敛的傅里叶积分的维纳代数。所得结果通过一些例子得到了补充。
英文摘要：
The following problem is studied in this paper: Which multipliers $\{λ_{k, n}\}$ ensure the convergence, as $n\to \infty$, of the linear means of the Fourier series of functions $f\in L_1[-π, π]$ $$ \sum_{k=-\infty}^\infty λ_{k, n}\hat{f}_k e^{ikx}, $$ where $\widehat{f}_k$ is the $k$-th Fourier coefficient, at a point at which the derivative of the function $\int_0^x f$ exists. A criterion for the convergence of the $(C, 1)$-means ($λ_{k, n}=(1-\frac {|k|}{n+1})_+$) is found, while in the general case $λ_{k, n}=ϕ(\frac {k}{n+1})$ a sufficient condition is derived for the convergence at all such points (that is, almost everywhere). The answer is given in terms of the belonging of $ϕ(x)$ and $xϕ'(x)$ to the Wiener algebra of absolutely convergent Fourier integrals. The obtained results are supplemented by some examples.
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