Landau常数和Lebesgue常数的渐近性与不等式  

Asymptotics and inequalities for the constants of Landau and Lebesgue

作　　者：陈超平 Chaoping Chen

机构地区：[1]河南理工大学数学与信息科学学院,焦作454000

出　　处：《中国科学：数学》2018年第7期923-938,共16页Scientia Sinica：Mathematica

摘　　要：对于所有的整数n≥0,Landau常数和Lebesgue常数分别定义为G_n=∑nk=01/16~k(2k/k)~2和L_n=1/2π∫_(-π)~π|sin((n+1/2)t)/sin(1/2t)|dt.本文给出G_n和L_(n/2)新的渐近级数.基于获得的结果,本文建立了Landau常数和Lebesgue常数新的不等式.设f∈C[-1,1],(s_nf)(x)=∑_(k=0)~na_kT_k(x)是f的Chebyshev展开式的部分和.Cheney指出,对于所有直到400为止的n值,当用最佳多项式逼近替代s_nf时,精度至多提高一位十进小数.本文证明了Cheney的论断对于n≤191833603亦真,而且本文说明了191833603不能被更大的整数替代.The constants of Landau and Lebesgue are defined, for all integers n≥0, respectively, by Gn=∑nk=01/16k（2k/k）2 and L_n=1/2π∫-ππ｜sin（（n＋1/2）t）/sin（1/2t）｜dt.In this paper, we give new asymptotic series for Gn and Ln/2. Based on the obtained results,we establish new inequalities for the constants of Landau and Lebesgue. Let f ∈ C[-1, 1], and let （snf）（x）=∑k=0nakTk（x）be the partial sum of the Chebyshev expansion of f. Cheney pointed out that, for all n up to 400, one can secure at most one extra decimal place of accuracy in replacing s_nf by the polynomial of best approximation. We prove that Cheney＇s claim is also true for n≤191833603. Moreover, we show that 191833603 cannot be replaced by a larger integer.

关 键 词：Landau常数 LEBESGUE常数 不等式 渐近展开式 

分 类 号：O178[理学—数学]