分母为正系数多项式的有理函数逼近的整体和点态估计

摘要

对任意定义在[0,1]上的非负连续函数f(x)(f≠0),该文证得:存在一个正系数多项式Pn(x)∈Ⅱn(+),使得|f(x)-1/Pn(x)∣≤Cωφλ(f,n-1/2 An1-λ(x)),其中An(x)=(√x(1-x))+1/(√n),0≤λ≤1,而Ⅱn(+)表示由所有次数不超过n的正系数多项式构成的集合.当f(x)在(0,1)内恰好改变l次符号时,该文构造了有理函数r(x)∈Rn(+)l,使得|f(x)-r(x)|≤C(l+1)2wφλ(f,n-1/2 An1-λ(x)).%For non-negative continuous function f (x) defined on [0, 1], and f(≡) 0, the present paper proves that, there is a polynomial Pn (x) ∈∏n(+) such that ｜f (x) - 1/Pn(x)｜ ≤ C (w)ψλ (f,n-1/2A1-λn(x)),where An(x) = (√x(1-x)) + 1/(√n), 0 ≤λ≤ 1, and ∏n(+) indicates the set of all polynomials of degree n with positive coefficients. When f (x) has exact l sign change points in (0, 1), we also construct a rational function r(x) ∈Rln (+) such that ｜f (x) - r(x)｜ ≤ C (l+1)2(w)ψλ (f,n-1/2A1-λn(x)).