在Freud正交多项式零点处的算子收敛性

The Convergence of Interpolation Operators Based on the Zeros of Orthogonal Polynomials with Freud Weights

设wβ(x)=e-12|x|β(β>76为Freud权,Freud正交多项式定义为关于上述定义的指数型Freud权正交的多项式,其零点分布在全实轴上。该文将Freud正交多项式零点作为插值结点,讨论了Hermite插值算子在全实轴上的收敛性,并得到:对实数轴上的任意一点X,Hermite算子收敛至函数f(x)。其中,yk=O(e(1/2-δ0)|xk|β),f(x)为实数轴上任一满足|f(x)|=O(e(1-ε0)|x|β)的连续函数。

Let Wβ(x)=e-12|x|β(β>76 be a certain type of Freud weights,{pn(x)}∞n=1 be the sequence of orthogonal polynomials with respect to the Freud weights,and the zeros of pn(x) are distributed on the whole real line.The present paper investigates the convergence of Hermite interpolator operators at the zeros of the orthogonal polynomials for the Freud weights.We prove that limn∞Hn(f,x)=f(x) holds for every x where yk=O(e(1/2-δ0)|xk|β),|f(x)|=O(e(1-ε0)|x|β).