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

TL;DR

This paper provides a complete characterization of almost sure convergence of orthogonal series using the existence of a majorizing measure, linking convergence to measure-theoretic conditions on the series coefficients.

Contribution

It introduces a new characterization of a.s. convergence of orthogonal series via majorizing measures, extending previous partial results.

Findings

Characterization of a.s. convergence in terms of majorizing measures

Conditions involving measure on a specific set T

Application of weakly majorizing measures and partitioning scheme

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.