导数在边界上具有Lipα条件的调和函数插值逼近阶

Order of Interpolatory Approximation to Harmonic Functions with u^((q))∈Lip α on the Boundary

D是由复平面z中一条Jordal闭曲线Γ围成的单连区域,z=0∈D.函数u(z)在D内调和且在Γ上u(q)∈L中α(0<α<1).基于复插值逼近理论证明了:存在唯一的调和插值多项式u_n~*(z),它与调和函数u(z)在Γ的摄动Fejer点{z_k~*}_0^(n-1)上有相同的值,在D上一致收敛于u(z),且收敛是稳定的.所得结果改进并推广了同类课题中已有的工作.

Let D be a simply connected domain in the complex z-plane bounded by a closed Jordan curve Γ,z = 0 ∈ D,and let the function u（z） be harmonic in D with u^（（q）） G Lip α（0 α 1） on Γ.Based on the theory of complex approximation by interpolation,it is proved that there exists a unique harmonic interpolation polynomial u_n~＊（z） which coincides with u（z） at disturbed Fejer points {z_k~＊}_0^（n-1） on T,and uniformly approximates to u（z） on D.The convergence is stable.The results obtained have improved or extended earlier similar works on this topic.