多项式的倒数对可微函数的逼近

APPROXIMATION OF DIFFERENTIABLE FUNCTIONS BY RECIPROCALS OF POLYNOMIALS

设非线性函数,f(x)∈C[-1,1]是非负的,f′(x)∈C[-1,1],f■(x)=f(x)+ε,其中ε<0,C■是与ε无关的常数,当,f(x)满足[f'(x)]~2/f_■(x)≤C■时,存在次数不超过n的代数多项式P_n(x),使得f(x)-1/P_n(x)1≤C_f~″·1/nω(f′,1/n)(C_f~■仅与C■有关)。根据这个定理,得到多项式f(x)=x^2或x_+~2的倒数的逼近阶是0(2/n^2)。

In this paper, we prove the following theorem: If nonlinear function f(x)∈C[-1, 1] be nonnegative, f'(x)∈C[-1, 1] and f_■(x)=f(x)+ε, where ε>0, if there exists a constant Cf which is independent of ε, when f(x) Satisfies the following result, we have algebraic polynomials P_n(x) of degree≤n, such that. Where constant C_f~' depends only on Cf.By this theorem, we obtain that the approximation degree of f(x)=x^2 or x_+~2 (x∈[-1, 1]) by reciprocals of polynomials is O (1/(n^2)).