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In the present paper we give a brief review of L 1 -convergence of trigonometric series. Previous known results in this direction are improved and generalized by establishing a new condition.

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By employing new ideas and techniques, we will refigure out the whole frame of L 1-approximation. First, except generalizing the coefficients from monotonicity to a wider condition, Logarithm Rest Bounded Variation condition, we will also drop the prior requirement fL 2π 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.

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To verify the universal validity of the ``two-sided'' monotonicity condition introduced in [4], we will apply it to include more classical examples. The present paper selects the L p convergence case for this purpose. Furthermore, Theorem 3 shows that our improvements are not trivial.

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