L_([0,1])~P(1＜p＜∞)空间正系数多项式的倒数逼近的Jackson型估计(英文)

The Jackson Estimation of Approximation by Reciprocals of Polynomials With Positive Coefficients in L_([0,1])~P Spaces for 1<p<∞

本文讨论L[0,1]^p（1〈p〈∞）空间函数的正系数多项式的倒数逼近Jackson型的估计问题，并证明了如下结论：设f（x）∈L[0,1]^p，1〈1〈p〈∞，且在（0，1）内改变l（l≥2）次符号，则存在0〈b1〈b2〈…〈b1〈1及一个n次多项式R（x）∈Πn（＋）使得｜｜f（x）-Πj=1^l（x-bj）/Pn（x）｜｜L[0,1]^p≤Cp，b,lw（f,n^-1/2）L[0,1]^p,其中Πn（＋）={pn（x）：pn（x）=Σ0≤k＋l≤n ak,lx^k（1-x）^l,ak,l≥0}为次数不超过n的正系数多项式的全体，b=min{｜bj＋1-bj｜：j=1，2，…，l-1），Cp,b,l表示与p，b及l有关的正常数．

The present paper investigates the Jackson estimation of approximation by reciprocals of polynomials with positive coefficients in L[0,1]^p spaces for 1 〈 p 〈∞ and proves P that： if f（x） ∈L[0,1]^p, 1 〈 p 〈 ∞, change its sign exactly 1 ≥ 2 times in （0, 1）, then there exist 0 〈 b1 〈 b2 〈 ... 〈 b1 〈 1 and a polynomial Pn（x） ∈ Πn（＋） such that ｜｜f（x）-Πj=1^l（x-bj）/Pn（x）｜｜L[0,1]^p≤Cp,b,lw（f,n^-1/2）L[0,1]^p, where Πn（＋） indicates all polynomials of degree n with positive coefficients, b = min{｜bj＋1 - bj｜ ： j = 1, 2,… , l - 1}, Cp,b,l is a positive constant only depending on p, b and l.