Ultimate Generalization to Monotonicity for Uniform Convergence of Trigonometric Series

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dc.creatorZhou, Song-Ping
dc.creatorZhou, Ping
dc.creatorYu, Dan-Sheng
dc.date2006-11-27
dc.date.accessioned2026-07-07T07:33:22Z
dc.date.available2026-07-07T07:33:22Z
dc.descriptionChaundy 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 Jolliffe theorem in the complex space.
dc.description21 pages
dc.identifierhttps://arxiv.org/abs/math/0611805
dc.identifierhttp://arxiv.org/abs/math/0611805
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/119430
dc.subjectClassical Analysis and ODEs
dc.subject42A20; 42A32
dc.titleUltimate Generalization to Monotonicity for Uniform Convergence of Trigonometric Series
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