(0,δ~M)三角插值多项式对函数及其导数的同时逼近

The(0,δ~M) trigonometric interpolation polynomials'simultaneous approximation of functions and their derivatives

证明了(0,δ~M)三角插值多项式L_(n,t)^((M))(f,x)的s(s=0,1,2,…,q)阶导数一致收敛于函数f(x)的s(s=0,1,2,…q)阶导数:设f(x)∈C_(2π),f(x)具有q阶连续导数,且f^((q))(x)∈Lipα,0<α<1,若β_k=O((|sin^M(nh)|)/n^(q+a))(k=0,1,2,…,n-1),则|[L_(n,t)^((M))(f,x)]^((s))-f^((x))(x)|=O((lnn)/n^(q-s+a))(s=0,1,2,…,q)。

It is proved in this paper that, the derivative of （0,δ^M）trigonometric interpolation polynomial Ln,ε^（M）（f,x）converges uniformly to f（s） （x）, （s=0,1,2,…,q）, if f（x）∈C2π,f（x） has qth continuous derivatives, and f（q（x）∈Lipα,0〈α〈1, if βk=O（｜sin^M（nh）｜/n^qα）（k=0,1,2,…,n-1）, then ｜Ln,ε^（M）（f,x）^（s）-f（s）（x）｜=O（1nn/n^qs＋α）（s=0,1,2,…,q）.