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2013 Higher-Order Hermite-Fejér Interpolation for Stieltjes Polynomials
Hee Sun Jung, Ryozi Sakai
J. Appl. Math. 2013: 1-15 (2013). DOI: 10.1155/2013/542653


Let wλ(x):=(1-x2)λ-1/2 and Pλ,n be the ultraspherical polynomials with respect to wλ(x). Then, we denote the Stieltjes polynomials Eλ,n+1 with respect to wλ(x) satisfying -11wλxPλ,nxEλ,n+1xxmdx =0, 0m<n+1; 0, m=n+1. In this paper, we consider the higher-order Hermite-Fejér interpolation operator Hn+1,m based on the zeros of Eλ,n+1 and the higher order extended Hermite-Fejér interpolation operator 2n+1,m based on the zeros of Eλ,n+1Pλ,n. When m is even, we show that Lebesgue constants of these interpolation operators are O(nmax{(1-λ)m-2,0})(0<λ<1) and Onmax1-2λm-2,00<λ<1/2, respectively; that is, 2n+1,m=O(nmax{(1-2λ)m-2,0})(0<λ<1) and Hn+1,m=Onmax1-λm-2,00<λ<1/2. In the case of the Hermite-Fejér interpolation polynomials 2n+1,m[·] for 1/2λ<1, we can prove the weighted uniform convergence. In addition, when m is 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 log n.


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Hee Sun Jung. Ryozi Sakai. "Higher-Order Hermite-Fejér Interpolation for Stieltjes Polynomials." J. Appl. Math. 2013 1 - 15, 2013.


Published: 2013
First available in Project Euclid: 14 March 2014

zbMATH: 06950739
MathSciNet: MR3138941
Digital Object Identifier: 10.1155/2013/542653

Rights: Copyright © 2013 Hindawi