关于正弦级数L^1-收敛性研究中对单调递减条件的确切推广  

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

作　　者：周颂平[1] 

机构地区：[1]浙江理工大学数学研究所,杭州310018

出　　处：《中国科学：数学》2010年第8期801-812,共12页Scientia Sinica：Mathematica

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 ＂computat

关 键 词：单调性 有界变差 正弦级数 收敛性 

分 类 号：O173[理学—数学]