# The Ultimate Condition to Generalize Monotonicity for Uniform Convergence of Trigonometric Series

Zhou, Songping and Zhou, Ping and Yu, Dansheng (2007) The Ultimate Condition to Generalize Monotonicity for Uniform Convergence of Trigonometric Series. [Preprint]

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## Abstract

Chaundy and Jolliffe [1] proved that if {a_n} is a non-increasing (monotonic) real sequence with lim_{n\rightarrow \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_{n\rightarrow \infty} n a_n = 0. We generalize (or weaken) the monotonic condition in this well-known 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.

Item Type: Preprint pubdom FALSE trigonometric series, uniform convergence, monotonicity, mean value bounded variation, AARMS 42-xx Fourier analysis > 42Axx Fourier analysis in one variable UNSPECIFIED lingyun ye 09 Jul 2007 19 Aug 2010 15:03 https://docserver.carma.newcastle.edu.au/id/eprint/366