L_p空间中最佳逼近的“集中”性质  

THE CONCENTRATION PHENOMENON OF THE BEST APPROXIMATION IN L_p SPACES

作　　者：李江波[1] 周颂平[2] 

机构地区：[1]丽水师范专科学校数学系,丽水323000 [2]浙江工程学院理学院,杭州310033

出　　处：《应用数学学报》2004年第2期265-273,共9页Acta Mathematicae Applicatae Sinica

基　　金：国家自然科学基金(10141001号);浙江省自然科学基金(101009号)

摘　　要：设 1≤p<+∞,f(x)是定义在[-1,1]上的 k阶可导且其 k阶导数p次幂可积的实函数,赋予通常的 L_p 范数,以∏_n表示次数不大于n的代数多项式的集合。本文发现了一类函数 f,在区间中某一固定内点 a具有性质其中常数 C,r与 n无关,这揭示了一个相当令人惊奇的现象,一些函数,例如第3节中提到的各幂函数,以及幂函数与“缓慢增长”函数的乘积函数,它们的 L_p 平均逼近特别是平均最佳逼近会“集中”在以某个内点为中心,长度为2r/n的小区间上。这就是我们称为的“集中”现象。Let 1≤p<+∞, and f(x) be a function whose kth derivative is p power intergrable over [-1, 1], with usual L_p norm. Denote by Ⅱ_n the class of polynomials of degree n. The present paper is devoted to the research on a class of functions f, which, for a fixed inner point a of the given interval, possesses the following property: ||f-P_n(f)||L_p[α-r/n,α+r/n]≥CE_n(f)_p, where C, r are constants independent of n, is a quite surprising phenomenon that the L_p mean approximation, especially mean best approximation, of some functions, such as power functions mentioned in §3, the products of power functions and 'slowly increasing' functions, can 'concertrate' in a small interval with an inner point as the center and r/n as the radius. This phenomenon is thus referred as the 'concertration phenomenon'.

关 键 词：Lp空间 最佳逼近 “集中”性质 实函数 LP范数 代数多项式 

分 类 号：O174[理学—数学]