Banach空间中无穷级数部份和的估计及应用

Estimations of partial Sum of the Vector Series on Banach Spaces and Its Application

本文定义了Banach空间中的囿变算子,提出了Banach空间无穷级数部份和∑T(n)的估计问题设T是从[1,+∞)到Banach空间E的一个囿变算子,本文的主要结果是:(1)指出了sum from i=1 to n T(i)-∫_1~nT(x)dx在Banach空间E中收敛;(2)给出了∑T(n)的两个估计。作为上述结论的应用,推广了实域R和复域C中一系列有关的经典结果。特别是文中关于Bernoulli数的新估计式,不仅可以简洁明快地解决并推广L.I.Nicolacscu[5]所提出的问题,而且可以给出不少有关的新结果。

The concept of bounded variation operator T: [ a, ∞)→X, where X is a Banach space, is introduced and the estimations of ∑T(n) are given.It is shown that ∑T(k)-∫1n T(x)dx converges to a point in X.The results generalize many classical results in real and complex fields [ 3,6and7 ] (e.g.Lim ) = c(r),r≠l,exists and 0≤c(r)≤l).The new estimation of Bernoulli's numbers, , is given [ 6 ] and the estimating problem proposed by L.I.Nicalaescu [ 5 ] is solved.