近乎Hermite-Fejér插值多项式之逼近

On the Approximation by Almost Hermite-Fejér Interpolating Polynomials

<正> §1引言 设C_[-1.1]是[-1,1]上连续函数之全体,C_[-1,1]~1是C_[-1,1]中连续可微函数所成之子集.对于,f∈C_[-1,1],记‖f‖为共上界范数,ω(f,δ)为共连续性模.设,J_(x)是阶为(1/2,-1/2)的n次Jacobi多项式。

Let{xk}k-1n be the zeros of the n-th Jacobi polynomial with weight (1 -x)1/2(1+ x)-1/2. For a continuous function f(x) on the interval [ - 1,1 ],there exists a unique algebraic polynomial Hn(f,x) with degree≤2n such that Hn(f,xk) =f(xk)(k= 0,1, … ,n,x0 = l) and Hn1(f,xk) =0 (k = 1,2, … ,n). Hn(f,x) is called almost Hermit-Fejer interpolating polynomial with nodes {xk}k-1n of function f(x).In this paper, we discuss the approximation of f(x) by Hn(f,x). First of all, we give the precise order of the quantity sup{|Hn(f,x) - f(x) |; ∈HW} and sup{ |Hn(f,x)-f(x) |;f'∈Hw},and point out that the sequence {Hn} is saturated, the saturationorder is { 1/n } and the saturation class is set of all f(x)such that f(cos θ) and f(cos θ)both belong Lip 1, Second, we give the asymptotic expansion of the quantity sup{|Hn(f,x)-f(x)|,f∈Lip11}.