Higher-Order Hermite-Fejér Interpolation for Stieltjes Polynomials

摘要

Letwλ(x):=(1-x2)λ-1/2andPλ,nbe the ultraspherical polynomials with respect towλ(x). Then, we denote the Stieltjes polynomialsEλ,n+1with respect towλ(x)satisfying∫-11‍wλxPλ,nxEλ,n+1xxmdx =0, 0≤m<n+1; ≠0, m=n+1. In this paper, we consider the higher-order Hermite-Fejér interpolation operatorHn+1,mbased on the zeros ofEλ,n+1and the higher order extended Hermite-Fejér interpolation operatorℋ2n+1,mbased on the zeros ofEλ,n+1Pλ,n. Whenmis even, we show that Lebesgue constants of these interpolation operators areO(nmax{(1-λ)m-2,0})(0<λ<1)andOnmax1-2λm-2,00<λ<1/2, respectively; that is,ℋ2n+1,m=O(nmax{(1-2λ)m-2,0})(0<λ<1)andHn+1,m=Onmax1-λm-2,00<λ<1/2. In the case of the Hermite-Fejér interpolation polynomialsℋ2n+1,m[·]for1/2≤λ<1, we can prove the weighted uniform convergence. In addition, whenmis odd, we will show that these interpolations diverge for a certain continuous function on[-1,1], proving that Lebesgue constants of these interpolation operators are similar or greater than logn.