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Journal of Applied Mathematics
Volume 2013, Article ID 542653, 15 pages
http://dx.doi.org/10.1155/2013/542653
Research Article

Higher-Order Hermite-Fejér Interpolation for Stieltjes Polynomials

1Department of Mathematics Education, Sungkyunkwan University, Seoul 110-745, Republic of Korea
2Department of Mathematics, Meijo University, Nagoya 468-8502, Japan

Received 5 July 2013; Accepted 22 August 2013

Academic Editor: Qiankun Song

Copyright © 2013 Hee Sun Jung and Ryozi Sakai. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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 .