{"paper":{"title":"The complete characterization of a.s. convergence of orthogonal series","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Witold Bednorz","submitted_at":"2013-03-19T10:59:09Z","abstract_excerpt":"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 i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1303.4547","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}