a. s的完整特征。正交级数的收敛
The complete characterization of a.s. convergence of orthogonal series
摘要：
本文证明了a.s.正交级数在优化测度存在下的收敛性这意味着对于给定的$(a_n)^{\infty}_{n=1}$，$a_n>0$，系列$\sum^{\infty}_{n=1}a_nφ_n$是a. e.对于每个标准正交序列$(φ_n)^{\infty}_{n=1}$收敛当且仅当存在\[T=\{0\}\cup\Biggl\{\sum^m_{n=1}a_n^2,m\geq 1\Biggr\}\]上的测量$m$，使得\[\sup_{t\in T}\int^{\sqrt{D(T)}}_0(m(B(t,r^2)))^{-{1}/{2}}\,dr<\infty,\]其中$D(T)=\sup_{s,t\in T}|s-t|$和$B(t,r)=\{s\in T:|s-t|\leq r\}$。所提出的方法是基于弱优化措施和一定的分区计划。
英文摘要：
In this paper we prove the complete characterization of a.s. convergence of orthogonal series in terms of existence of a majorizing measure. It means that for a given $(a_n)^{\infty}_{n=1}$, $a_n>0$, series $\sum^{\infty}_{n=1}a_nφ_n$ is a.e. convergent for each orthonormal sequence $(φ_n)^{\infty}_{n=1}$ if and only if there exists a measure $m$ on \[T=\{0\}\cup\Biggl\{\sum^m_{n=1}a_n^2,m\geq 1\Biggr\}\] such that \[\sup_{t\in T}\int^{\sqrt{D(T)}}_0(m(B(t,r^2)))^{-{1}/{2}}\,dr<\infty,\] where $D(T)=\sup_{s,t\in T}|s-t|$ and $B(t,r)=\{s\in T:|s-t|\leq r\}$. The presented approach is based on weakly majorizing measures and a certain partitioning scheme.
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。
