Jordan区域上广义Lagrnage插值多项式的平均逼近阶

Order of Approximation in the Mean by Generalized Lagrange Interpolating Polynomials in Jordam Domain

本文在对Jordan区域D的边界加上较弱的光滑性条件下,考虑函数f(z)∈E(D),P>1,在Fejer插值点上的广义Lagrange插值多项式L_N(f,z)(见公式(1.5)),得到了平均逼近阶为ω(f,1/n)_p—函数f(z)在L^p((?)D)意义下的连续模在1/n处的值,阐明了用函数f(z)∈A(D)的Lagrage插值多项进行逼近时,是不可能得到这样的逼近阶的。

The generalized Lagrange interpolating polynomials for f(Z)∈E(D), P>1 at the Fejer interpolation points are studied.Under certain mild condition on the smoothness of the boundry curve we give the order of appcoximation in themean as wi (f,1/n)_p—the modulus of continuity of f(z) in L^P(Γ) norm. It isexplained that it is impossible for the Lagrange interpolating polynomials.