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  is disproved in Section 3.
 Kórus, P.2010Remarks on the uniform and L1-convergence of trigonometric seriesActa Math. Hungar.128369–380.