{"paper":{"title":"Necessary and sufficient Tauberian conditions for the logarithmic summability of functions and sequences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Ferenc Moricz","submitted_at":"2012-06-27T07:16:04Z","abstract_excerpt":"Let $s: [1, \\infty) \\to \\C$ be a locally integrable function in Lebesgue's sense on the infinite interval $[1, \\infty)$. We say that $s$ is summable $(L, 1)$ if there exists some $A\\in \\C$ such that $$\\lim_{t\\to \\infty} \\tau(t) = A, \\quad {\\rm where} \\quad \\tau(t):= {1\\over \\log t} \\int^t_1 {s(u) \\over u} du.\\leqno(*)$$ It is clear that if the ordinary limit $s(t) \\to A$ exists, then the limit $\\tau(t) \\to A$ also exists as $t\\to \\infty$. We present sufficient conditions, which are also necessary in order that the converse implication hold true. As corollaries, we obtain so-called Tauberian th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.6188","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"}