Ultimate Generalization to Monotonicity for Uniform Convergence of Trigonometric Series

Details Ultimate Generalization to Monotonicity for Uniform Convergence of Trigonometric Series Ultimate Generalization to Monotonicity for Uniform Convergence of Trigonometric Series

2006-11-27

arxiv.org/abs/math/0611805v1

Mathematics

Classical Analysis and ODEs

21 pages

Scientific paper

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 Jolliffe theorem in the complex space.

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