【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。更多还原

【Abstract】 Let 1≤p<+∞, and f(x) be a function whose kth derivative is p powerintergrable over [-1, 1], with usual L_p norm. Denote by Ⅱ_n the class of polynomials ofdegree n. The present paper is devoted to the research on a class of functions f, which, fora 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 bestapproximation, of some functions, such as power functions mentioned in §3, the productsof power functions and "slowly increasing" functions, can "concertrate" in a small intervalwith an inner point as the center and r/n as the radius. This phenomenon is thus referredas the "concertration phenomenon".更多还原