插值逼近具边界数据的调和函数

该文考虑光滑闭Jondan曲线Γ围成的单连区域D,证明了在Γ上具有已知导数数据的D内调和函数u(x,y)的存在性.继而构造了一个调和插值多项式序列在(?)=D∪Γ上一致收敛于u(x,y),且具理想的收敛速度.此外,以往同类研究工作中的边界Γ是解析曲线,而在该文中已减少边界限制为Γ∈J_0.

Interpolatory Approximation to Harmonic Function with Boundary Data

Suppose D is a simply connected domain bounded by a smooth closed Jordan curve Γ. In D the existence of a  harmonic function u(x, y) with given derivative boundary data on Γ is proved. By the way, the line integral representation of such a harmonic function u(x, y) is obtained. Moreover, the authors construct a sequence of harmonic interpolation polynomials uniformly convergent to u(x, y) on D=D ∪ Γ with the desirable rate of  convergence. In addition, the boundary condition that Γ is an analytic curve in early similar works is decreased to Γ ∈ J₀.