Hermite-Fejér多项式逼近函数类Lipa,0<α≤1的展式

The Expansion for the Degree of Approximation of Function Class Lipα,o<α≤1 ,by Hermite-Fejér Polynomials

设三角矩阵{χ_k^(n)},κ=1,2,…,的第n行为t1次(?)多项式T_n(χ)=cos(n arc cos x)的根χ_k≡χ_k^(n)=cosθ_k=cos2κ-1/2nπ,κ=1,2,…。

In this paper the expansion for the degree of approximation of function class Lipα(0<α≤1)by Hermite-Fejer interpolation polynomials at the Chebyshev nods{ cos((2k-1)π/2n), k = 1,2,… ,n}is discussed.It is proved that the Δj(η) = Hn (f1,cos(jπ/n+η))are strictly increasing function of j,j = 0,1 ,…, [2/n] , where f1(t) = |x -t|α.For Hn{Lipα}=sup{‖Hn(f,·)-f(·)‖:f∈Lipα}, 0<α<l, when estimation of upper and lower bounds is made the expansion with three terms is obtained.Finally, the expansion of Hn {Lip1} is studied, where the case n = 2m+1, m = l,2, - is not pointed out in[3].