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

Witold Bednorz

• 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 (an)∞n=1, an>0, series ∑∞n=1anφn is a.e. convergent for each orthonormal sequence (φn)∞n=1 if and only if there exists a measure m on T={0}∪{∑n=1ma2n,m≥1} such that supt∈T∫D(T)√0(m(B(t,r2)))−1/2dr<∞, where D(T)=sups,t∈T|s−t| and B(t,r)={s∈T : |s−t|≤r}. The presented approach is based on weakly majorizing measures and a certain partitioning scheme.