{"paper":{"title":"Abel Continuity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Huseyin Cakalli, Mehmet Albayrak","submitted_at":"2011-01-07T14:15:51Z","abstract_excerpt":"A sequence $\\textbf{p}=(p_{n})$ of real numbers is called Abel convergent to $\\ell$ if the series $\\Sigma_{k=0}^{\\infty}p_{k}x^{k}$ is convergent for $0\\leq x<1$ and \\[\\lim_{x \\to 1^{-}}(1-x) \\sum_{k=0}^{\\infty}p_{k}x^{k}=\\ell.\\] We introduce a concept of Abel continuity in the sense that a function $f$ defined on a subset of $\\Re$, the set of real numbers, is Abel continuous if it preserves Abel convergent sequences, i.e. $(f(p_{n}))$ is an Abel convergent sequence whenever $(p_{n})$ is. A new type of compactness, namely Abel sequential compactness is also introduced, and interesting theorems"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1101.1440","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"}