What condition can correctly generalize monotonicity in L~1-convergence of sine series

In classical Fourier analysis, the integrability of trigonometric series is considered as an interesting but difficult topic. In particular, the integrability of sine series have not been touched in dozens years since Boas, Heywood published their classical results, meanwhile the generalizations of (deceasing) monotonicity have been developed from various quasimonotonicity and bounded variation conditions, finally, to the mean value bounded variation condition, an essential ultimate condition in most sense, and applied to various convergence problems extensively including uniform convergence, mean convergence, L p integrability and best approximation etc. The difficulty of the research can be seen from this point. We may need another point of view now. Given a sine series ∞ n=1 a n sin nx, its sum function can be written as g(x) at the point x where it converges. However, it is usually a very hard job to verify if the sum function or the sine series itself belonging to L 2π or not. On the other hand, in studying L 1 -convergence problems, people usually need a requirement that g ∈ L 2π , which also becomes a hard condition to check or a prior condition to set in most cases. For instance, the well-known classical results for L 1 -convergence says that, let the real even (odd) function f ∈ L 2π , and its Fourier cosine (sine) coefficients {a n } ∞ n=1 ∈ MS, then lim n→∞ f - S n (f) L 1 = 0 if and only if lim n→∞ a n log n = 0. On the other hand, the classical result of uniform convergence of sine series (Chaudy-Jolliffe theorem) says that, let {a n } ∞ n=1 ∈ MS, then the sine series ∞ n=1 a n sin nx uniformly converges if and only if lim n→∞ na n = 0. The difference of two types of classical theorems is very clear: one need a prior condition f ∈ L 2π , the other does not require that f ∈ C 2π . Mathematicians surely prefer the latter to the former in mathematical sense. The reason that the prior condition in L 1 case cannot be avoided mainly arises from the much more "computation complexity " in the integrable space than the continuous space, and technical benefit to turn the computation from "infinity" into "finiteness" by setting the integrability. Furthermore, people also note that the first important problem in the integrable space should be L 1 -convergence, which may be achieved by the series itself (by coefficients) without mentioning the sum function. Based upon these reasons, why do we not try to find a more direct and clean version of L 1 -convergence? The present paper is arranged as follows. First construct a nonnegative sequence which shows that although it is quasimonotone and the inequality ∞ n=1 n -1 a n ∞ holds, but the corresponding sine series does not converge in L 1 -norm (Theorem 1). Then, we raised a new correct condition which can guarantee the above mentioned condition to be necessary and sufficient for which the corresponding sine series does converge in L 1 - norm (Theorem 2), and also discuss the relationships with other related sets (Theorems 3 and 4). Finally, we prove the newly raised condition cannot be weakened further in this L 1 -convergence case in some sense (Theorem 5). As a whole, we give a complete solution to this topic.