{"paper":{"title":"Ultimate Generalization to Monotonicity for Uniform Convergence of Trigonometric Series","license":"","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Dan-Sheng Yu, Ping Zhou, Song-Ping Zhou","submitted_at":"2006-11-27T02:07:06Z","abstract_excerpt":"Chaundy and Jolliffe [4] proved that if $\\{a_{n}\\}$ is a non-increasing (monotonic) real sequence with $\\lim\\limits_{n\\to \\infty}a_{n}=0$, then a necessary and sufficient condition for the uniform convergence of the series $\\sum_{n=1}^{\\infty}a_{n}\\sin nx$ is $ \\lim\\limits_{n\\to \\infty}na_{n}=0$. We generalize (or weaken) the monotonic condition on the coefficient sequence $\\{a_{n}\\}$ in this classical result to the so-called mean value bounded variation condition and prove that the generalized condition cannot be weakened further. We also establish an analogue to the generalized Chaundy and J"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0611805","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/math/0611805/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}