By employing new ideas and techniques, we will refigure out the whole frame of L1-approximation. First, except generalizing the coefficients from monotonicity to a wider condition, Logarithm Rest Bounded Variation condition, we will also drop the prior requirement f∊L2π but directly consider the sine or cosine series. Secondly, to achieve nontrivial generalizations in complex spaces, we use a one-sided condition with some kind of balance conditions. In addition, a conjecture raised in [9] is disproved in Section 3.
[1] Kórus, P. 2010 Remarks on the uniform and L1-convergence of trigonometric series Acta Math. Hungar. 128 369–380 .
[2] Le, R. J., Zhou, S. P. 2007 On L1 convergence of Fourier series of complex valued functions Studia Sci. Math. Hungar. 44 35–47.
[3] Leindler, L. 2001 On the uniform convergence and boundedness of a certain class of sine series Anal. Math. 27 279–285 .
[4] Tikhonov, S. 2008 On L1-convergence of Fourier series J. Math. Anal. Appl. 347 416–427 .
[5] Xie, T. F., Zhou, S. P. 1996 L1-approximation of Fourier series of complex valued functions Proc. Royal Soc. Edinburgh 126A 343–353 .
[6] Yu, D. S., Zhou, P., Zhou, S. P. 2009 On L1-convergence of Fourier series under the MVBV condition Canad. Math. Bull. 52 627–636 .
[7] Zhou, G. Z. 1998 Some remarks on L1-approximation J. Hangzhou Univ. Nat. Ed. 25 19–25 (in Chinese).
[8] Zhou, S. P. 2010 What condition can correctly generalize monotonicity in L1-convergence of sine series? Science China Math., Chinese Ed. 40 801–812 (in Chinese).
[9] Wang, M. Z., Zhou, S. P. 2010 Applications of MVBV condition in L1 integrability Acta Math. Hungar. 129 70–80 .