{"paper":{"title":"Statistical extension of classical Tauberian theorems in the case of logarithmic summability of locally integrable functions on $[1,\\infty)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Ferenc Moricz, Zoltan Nemeth","submitted_at":"2012-06-29T08:39:39Z","abstract_excerpt":"Let $s:[1,\\infty) \\to \\C $ be a locally integrable function in Lebesgue's sense. The logarithmic (also called harmonic) mean of the function $s$ is defined by [\\tau(t) := \\frac 1{\\log t} \\int_1^t \\frac {s(x)}{x} dx, \\qquad t>1,] where the logarithm is to base $e$. Besides the ordinary limit $\\lim_{x\\to \\infty} s(x)$, we also use the notion of the so-called statistical limit of $s$ at $\\infty$, in notation: $ \\stlim_{x\\to \\infty} s(x)=\\ell $, by which we mean that for every $\\e>0$, [\\lim_{b\\to \\infty} \\frac 1b \\Big | \\Big {x\\in(1,b): |s(x)-\\ell| >\\e \\Big} \\Big| = 0.] We also use the ordinary li"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.6963","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"}