The complete characterization of a.s. convergence of orthogonal series

Abstract

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 $(a_n{)}_{n=1}^{\infty }$, $a_n>0$, series ${\sum }_{n=1}^{\infty }{a}_n{\phi }_n$ is a.e. convergent for each orthonormal sequence $({\phi }_n{)}_{n=1}^{\infty }$ 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 $B(t,r)=\{s\in T:|s-t|\le r\}$. The presented approach is based on weakly majorizing measures and a certain partitioning scheme.

Published in at http://dx.doi.org/10.1214/11-AOP712 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)