On L1-Convergence of Fourier Series Under MVBV Condition

Yu, Dansheng and Zhou, Ping and Zhou, Songping (2006) On L1-Convergence of Fourier Series Under MVBV Condition. [Preprint]

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    Let f \in L_{2\pi} be a real-valued even function with its Fourier series a0/2 + \sum_{n=1}^\infty a_n \cos nx, and let Sn (f, x), n \geq 1, be the n-th partial sum of the Fourier series. It is well-known that if the nonnegative sequence {a_n} is decreasing and lim_{n\rightarrow \infty}a_n = 0, then lim_{n\rightarrow \infty} ||f− S_n(f)||_L = 0 if and only if lim_{n\rightarrow \infty}a_n log n = 0. We weaken the monotone condition in this classical result to the so-called mean value bounded variation (MVBV) condition. The generalization of the above classical result in real-valued function space is presented as a special case of the main result in this paper which gives the L1-convergence of a function f \in L_{2\pi} in complex space. We also give results on L1-approximation of a function f \in L_{2\pi} under the MVBV condition.

    Item Type: Preprint
    Additional Information: pubdom FALSE
    Uncontrolled Keywords: AARMS
    Subjects: 41-xx Approximations and expansions > 41Axx Approximations and expansions
    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 15:05

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