Journal of Inequalities and Applications
Volume 2010 (2010), Article ID 816363, 29 pages
doi:10.1155/2010/816363
Research Article

Derivatives of Orthonormal Polynomials and Coefficients of Hermite-Fejér Interpolation Polynomials with Exponential-Type Weights

H. S. Jung1 and R. Sakai2

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

Received 10 November 2009; Accepted 14 January 2010

Academic Editor: Vijay Gupta

Copyright © 2010 H. S. Jung and R. 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 ℝ=(−∞,∞), and let Q∈C2:ℝ→[0,∞) be an even function. In this paper, we consider the exponential-type weights wρ(x)=|x|ρexp(−Q(x)),  ρ>−1/2,  x∈ℝ, and the orthonormal polynomials pn(wρ2;x) of degree n with respect to wρ(x). So, we obtain a certain differential equation of higher order with respect to pn(wρ2;x) and we estimate the higher-order derivatives of pn(wρ2;x) and the coefficients of the higher-order Hermite-Fejér interpolation polynomial based at the zeros of pn(wρ2;x).