Higher-Order Hermite-Fejér Interpolation for Stieltjes Polynomials

• Main Authors: Hee Sun Jung, Ryozi Sakai
• Format: Article
• Language: English
• Published: Hindawi Limited 2013-01-01
• Series: Journal of Applied Mathematics
• Online Access: http://dx.doi.org/10.1155/2013/542653

Let and be the ultraspherical polynomials with respect to . Then, we denote the Stieltjes polynomials with respect to satisfying . In this paper, we consider the higher-order Hermite-Fejér interpolation operator based on the zeros of and the higher order extended Hermite-Fejér interpolation operator based on the zeros of . When is even, we show that Lebesgue constants of these interpolation operators are and , respectively; that is, and . In the case of the Hermite-Fejér interpolation polynomials for , we can prove the weighted uniform convergence. In addition, when is odd, we will show that these interpolations diverge for a certain continuous function on , proving that Lebesgue constants of these interpolation operators are similar or greater than log .