THE COMPLETE CHARACTERIZATION OF AS CONVERGENCE OF ORTHOGONAL SERIES

• 成果类型: Article
• 署名作者: Bednorz, Witold
• 署名单位: University of Warsaw
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/11-AOP712
• 发表日期: 2013
• 页码: 1055-1071
• 关键词: regularity; THEOREM

摘要

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)(infinity), a(n) > 0, series Sigma(infinity)(n=1) a(n)phi(n) is a.e. convergent for each orthonormal sequence (phi(n))(n=1)(infinity) if and only if there exists a measure m on T = {0} boolean OR {Sigma(m)(n=1) a(n)(2), m >= 1} such that sup(t is an element of T)integral(root D(T))(0) (m(B(t, r(2))))(-1/2) dr < infinity, where D(T) = sup(s,t is an element of T) vertical bar s - t vertical bar and B(t, r) = {s is an element of T :vertical bar s - t vertical bar <= r}. The presented approach is based on weakly majorizing measures and a certain partitioning scheme.