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  • 1 Institute of Mathematics, Zhejiang Sci-Tech University, Hangzhou, Zhejiang, 310018, P. R. China
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Let fL2π be a real-valued even function with its Fourier series , and let Sn(f,x) be the nth partial sum of the Fourier series, n≧1. The classical result says that if the nonnegative sequence {an} is decreasing and , then if and only if . Later, the monotonicity condition set on {an} is essentially generalized to MVBV (Mean Value Bounded Variation) condition. Very recently, Kórus further generalized the condition in the classical result to the so-called GM7 condition in real space. In this paper, we give a complete generalization to the complex space.

  • [1] Kórus, P. 2010 Remarks on the uniform and L1-convergence of trigonometric series Acta Math. Hungar. 128 369380 .

  • [2] Tikhonov, S. 2008 On L1-convergence of Fourier series J. Math. Anal. Appl. 347 416427 .

  • [3] Xie, T. F., Zhou, S. P. 1996 L1-approximation of Fourier series of complex valued functions Proc. Royal Soc. Edinburg Sect. A 126 343353.

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  • [4] Yu, D. S., Zhou, P., Zhou, S. P. 2009 On L1-convergence of Fourier series under MVBV condition Canad. Math. Bull. 52 627636 .

  • [5] Zhou, S. P., A remark on L1-convergence of Fourier series under MVBV condition, Adv. Math. (Beijing), to appear.

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