非线性逼近的强唯一性  

STRONG UNIQUENESS OF NONLINEAR APPROXIMATION

作　　者：郭元明[1] 

机构地区：[1]湖南教育学院

出　　处：《数学杂志》1991年第4期387-391,共5页Journal of Mathematics

摘　　要：设G是赋范线性空间 E 的子集,x∈E,g_0∈G.称 g_0是 G 对 x 是 q(q>1)阶强唯一最佳逼近乃指:若存在常数 C=C(x)>0,满足‖x-g‖~q≥‖x-g_0‖~q+c‖g-g_0‖~q,(?)g∈G.(1)我们对所有满足(1)式的常数 C 上确界为 x 相对于 G 的强唯一常数,记作 r(x).本文先获得:若 G 是 L_p 或 H^(k,p)(K≥0,2≤p≤∞)的弱拟凸(伪凸、拟凸)集,则 G 对 x∈L_p(H^(k,p))的最佳逼近具有 p 阶强唯一性;然后在一般赋范空间证明了 r(x)相对于 x 是上半连续的.Let G be a subset of a normed linear space E and x∈E.An element g_o∈G is said to be strongly unique best approximation of order q(q>1)to x ifthere exists c=c(x)>0 such that‖x-g‖~q≥‖x-g_0‖~q+c‖g-g_0‖~q,(?)g∈G.(1)The supremum of all constants such that (1) is called the strong unicityconstant and is denoted by r(x).First,it is shown that if G is a weakly quasi-convex(pseudoconvex,quasiconvex) set of L_p or H^(k,p)(K≥0,2≤p<∞),thenthe best approximation to x∈L_p(H^(k,p))from G is strongly unique of order p.Second,it is proved that r(x)is semicontinuous with respect to x in a normedlinear space.

关 键 词：非线性逼近 强唯一性 弱拟凸集 

分 类 号：O174.41[理学—数学]