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 $(a_{n})^{\infty}_{n=1}$, $a_{n}>0$, series $\sum^{\infty}_{n=1}a_{n}\varphi_{n}$ is a.e. convergent for each orthonormal sequence $(\varphi_{n})^{\infty}_{n=1}$ 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.