样条子空间逼近周期可微函数类的最佳逼近度

The Best Approximation of Periodic Differentiable Class Approximated by Spline Subspace

样条函数类与周期函数类的逼近问题是现代逼近论研究中的热点问题之一.本文引入r阶样条子空间SrΔN、周期可微函数类Lmp、函数类WpmSrΔN和函数类WprΔN,运用对偶性原理和连续模概念,研究了用SrΔN逼近Lpm的最佳逼近度问题,得出了其最佳逼近上确界:当f∈Lqm∩Lp时有E(f,SrΔN)p≤E(f(m),Sr-ΔNm)qsupg∈Wmp′(SrΔN)‖g‖q′.同时,也研究了函数类WpmSrΔN与函数类WrpΔN之间的关系,得出了当f∈Lq(1≤q<∞)和f∈C时的最佳逼近结果.

The approximation of spline function and periodic function classes is important in research of modern approximation. In this paper,spline subspace S^r△N with degree r, periodic differentiable class Lp^m, class Wp^m （S^r△N ） and Wp^r （△N） are introduced. Using the duality principle, the best approximation of Lp^m approximated by S^r△N is studied, and the best approximation supermum is obtained, that＇s,for arbitary f∈ Lq^m ∩Lp ,E（f,S^r△N ）,≤E（f^（m） ,S△N^r-m）, sup g∈Wp^m（S^r△N） ｜｜ g ｜｜ q′ holds. Meanwhile,the relation between class Wp^m （S^r△N） and Wp^r （△N） is studied, and the best approximations for f∈Lq （1≤q〈∞） and f∈C are obtained.