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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
    Additional Information: pubdom FALSE
    Uncontrolled Keywords: trigonometric series, uniform convergence, monotonicity, mean value bounded variation, AARMS
    Subjects: 42-xx Fourier analysis > 42Axx Fourier analysis in one variable
    Faculty: UNSPECIFIED
    Depositing User: lingyun ye
    Date Deposited: 09 Jul 2007
    Last Modified: 19 Aug 2010 15:03
    URI: https://docserver.carma.newcastle.edu.au/id/eprint/366

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