关于Lebesgue点处Cesàro均值的Belinsky猜想的一个反例

中文摘要 AI 辅助

1997年，Belinsky猜想：对于凸子序列，Carleson、Trigub和Zagorodny的对数增长条件是子序列傅里叶部分和的算术均值在每个可积函数的每个Lebesgue点收敛的充要条件。我们反驳该猜想的充分性部分，具体而言，我们构造了一个严格凸递增序列$(a_m)$满足$a_m\leq 7m^8$，以及一个函数$f\in L^1(\mathbb T)$，其中0是Lebesgue点，$f(0)=0$，且均值$m^{-1}\sum_{k=1}^m S_{a_k}f(0)$无界。

英文摘要

In 1997, Belinsky conjectured that, for convex subsequences, the logarithmic growth condition of Carleson, Trigub, and Zagorodni\uı is necessary and sufficient for the arithmetic means of subsequential Fourier partial sums to converge at every Lebesgue point of every integrable function. We disprove the sufficiency part of this conjecture. More precisely, we construct a strictly convex increasing sequence $(a_m)$ satisfying $a_m\leq 7m^8$ and a function $f\in L^1(\mathbb T)$ for which $0$ is a Lebesgue point, $f(0)=0$, and the means $m^{-1}\sum_{k=1}^m S_{a_k}f(0)$ are unbounded.

发表机构

• United Arab Emirates University（阿联酋大学）

机构由 AI 辅助整理，请以论文原文为准。