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

• Autores: Witold Bednorz
• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 41, Nº. 2, 2013, págs. 1055-1071
• Idioma: inglés
• DOI: 10.1214/11-AOP712
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• Resumen

  • 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.