关于Turán的一个未解决的问题

On Turan's an Open Problem

本文讨论了以盖根堡多项武C_n^(λ)(x)的零点{x_k^(λ)}_k^n=1为基点的拟Hermite—Fejer插值多项式E_n^(λ)(f,x)的收敛性问题,证明当0≤λ≤1/2时,E_n^(λ)(f,x)在闭区间[-1,1]上一致收敛于连续函数f(x),部分地解决了P.Turan提出的一个问题。

It is discussed that the convergence problem of the Quasi—Hermite—Fejer interpolation polynomial E_n^(λ)(f, x) based on the zeros {x_k^(λ)}_(k=1)~n of Gegenbauer polynomial C_n^(λ)(x). It is proved that if 0≤λ≤1/2, then E_n^(λ)(f, x) converges uniformly to the continuous function f(x) in closed interval [-1, 1]. This solved partly a question propounded by Turan.