摘要
本文在处理L1-收敛性问题中给出了一个确切的条件和一种更直接的方式.
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, Lp 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 an sinnx, 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 L2π or not. On the other hand, in studying Ll-convergence problems, people usually need a requirement that g ∈ L2π, 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 Ll-convergence says that, let the real even （odd） function f∈ L2π, and its Fourier cosine （sine） coefficients {an}n∞=l E MS, then limn→∞ ｜｜f - Sn（f） ｜｜L1 = 0 if and only if lim~ an logn = 0. On the other hand, the classical result of uniform convergence of sine series （Chaudy-Jolliffe theorem） says that, let { an}n∞=1 ∈ MS, then the sine series ∑=1 an sinnx uniforvnly converges if and only if limn→∞ nan =0. The difference of two types of classical theorems is very clear： one need a prior condition f ∈ L2π, the other does not require that f ∈ C2π. Mathematicians surely prefer the latter to the former in mathematical sense. The reason that the prior condition in L1 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 LLconvergence, which may be achieved by the series itself （by coefficients） without mentioning the sum function. Based upon these reasons, why do we The present paper is arranged as follows not try to find a more direct and clean version of L1-convergence？ First construct a nonnegative sequence which shows that although it is qnasimonotone and the inequality ∑n∞=1 n-1an 〈∞ holds, but the corresponding sine series does not converge in L1-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 L1- 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 L1-convergence case in some sense （Theorem 5）. As a whole, we give a complete solution to this topic.
出处
《中国科学：数学》
CSCD
北大核心
2010年第8期801-812,共12页
Scientia Sinica：Mathematica
关键词
单调性
有界变差
正弦级数
收敛性
trigonometric series, convergence, integrability, monotonicity
