Table of Contents
ISRN Mathematical Analysis
Volume 2012 (2012), Article ID 904169, 31 pages
http://dx.doi.org/10.5402/2012/904169
Research Article

Convergence and Divergence of Higher-Order Hermite or Hermite-Fejér Interpolation Polynomials with Exponential-Type Weights

1Department of Mathematics Education, Sungkyunkwan University, Seoul 110-745, Republic of Korea
2Science Division, Center for General Education, Aichi Institute of Technology, Yakusa-cho, Toyota 470-0392, Japan
3Department of Mathematics, Meijo University, Nagoya 468-8502, Japan

Received 30 December 2011; Accepted 26 February 2012

Academic Editors: B. Ricceri and W. Yu

Copyright © 2012 Hee Sun Jung et al. 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 = ( βˆ’ ∞ , ∞ ) , and let 𝑀 𝜌 ( π‘₯ ) = | π‘₯ | 𝜌 𝑒 βˆ’ 𝑄 ( π‘₯ ) , where 𝜌 > βˆ’ 1 / 2 and 𝑄 ∈ 𝐢 1 ( ℝ ) ∢ ℝ β†’ ℝ + = [ 0 , ∞ ) is an even function. Then we can construct the orthonormal polynomials 𝑝 𝑛 ( 𝑀 2 𝜌 ; π‘₯ ) of degree 𝑛 for 𝑀 2 𝜌 ( π‘₯ ) . In this paper for an even integer 𝜈 β‰₯ 2 we investigate the convergence theorems with respect to the higher-order Hermite and Hermite-Fejér interpolation polynomials and related approximation process based at the zeros { π‘₯ π‘˜ , 𝑛 , 𝜌 } 𝑛 π‘˜ = 1 of 𝑝 𝑛 ( 𝑀 2 𝜌 ; π‘₯ ) . Moreover, for an odd integer 𝜈 β‰₯ 1 , we give a certain divergence theorem with respect to the higher-order Hermite-Fejér interpolation polynomials based at the zeros { π‘₯ π‘˜ , 𝑛 , 𝜌 } 𝑛 π‘˜ = 1 of 𝑝 𝑛 ( 𝑀 2 𝜌 ; π‘₯ ) .