广义Bernoulli函数用三角多项式的单边逼近

ONE-SIDE'S APPROXIMATION OF THE GENERALIZED BERNOULLI'S FUNCTION BY TRIGONOMETRIC POLYNOMIAL

给出了广义Bernoulli函数用三角多项式在L_1范数下的最佳单边逼近的准确值。

Let G_r(t)=(1/2л)sum from v=(-∞) to (+∞) ((e^(ivt))/(p_r(it)))(i=(-1)^(1/2))be the generalized Bernoullis function,where P_r (λ)=λ~r+α_1λ^(r-1)+…+α_(n-1)λ+α_n=multiply form s=1 to k(λ~2-2α_sλ+α_s^2+β_s^2) multiply from j=1 to(r-2k)(λ-λ_j) .The following theorem is established: 1) Letλ=1,λ_1≠0,then there is N∈N such that for every n≥N, E_n^+(G_1)_(L_1)=-2л[_n(G_1,0)-1/2лλ_1] E_n^-(G_1)_(L_1)=-2л[_n(G_1,0)+1/2лλ_1]. 2)If λ≥2,then there is N∈N such that for erery n≥N E_n^+(G_r)_(L_1)=‖max_n(G_r,n)-_n(G_r,·)‖L_1, E_n^-(G_r)_(L_1)=‖_n(G_r,·)-min_n (Gr,n)‖L_1.