复插值逼近从单位圆盘扩大到Jordan区域的边界充要条件

D是复平面上Jordan闭曲线Γ围成的单连区域.自20世纪40年代以来,在闭域D上用Hermite插值多项式一致逼近和平均逼近f(z)∈A(q)(D),已经有很多研究而且还在继续研究.大量作者接连不断地减少区域边界条件以期提高论文的水平.什么是最少的边界条件?直到现在,这个问题似乎还没有解决.本文已经发现并证明了:为了Hermite插值多项式收敛于f(z)∈A(q)(D—),其充分必要条件是Γ∈C~,即设这区域边界曲线Γ的参数表示是z=z(t),它具有一阶导数z'(t)连续且不等于零于0,2π]上.

The Border Necessary and Sufficient Conditions of Complex Interpolation Approximation from the Unit Disk Widely Extended to Jordan Area

Suppose that D is a simply-connected domain bounded by a closed Jordan curve Γ in the complex plane. Since the 1940 s,lot of authors have been researching uniform approximation and approximation in mean to by Hermite interpolation polynomial. The authors successively reduce the domain boundary condition to improve the level of paper. What are the least boundary conditions？ Up to now,the problem does not seem to be resolved. In this paper,it is discovered and proved that for the approximation to f（z）∈A^（q）（^-D）by Hermite interpolation,the sufficient and necessary condition is Γ∈ ^～C,that is,the parameter representation of Γ is z =z（t） with z＇（t） continuous and non-vanishing on α,β].