Journal of Applied Mathematics

Asymptotic Properties of Derivatives of the Stieltjes Polynomials

Hee Sun Jung and Ryozi Sakai

Full-text: Open access

Abstract

Let w λ ( x ) : = ( 1 x 2 ) λ 1 / 2 and P λ , n ( x ) be the ultraspherical polynomials with respect to w λ ( x ) . Then, we denote the Stieltjes polynomials with respect to w λ ( x ) by E λ , n + 1 ( x ) satisfying 1 1 w λ ( x ) P λ , n ( x ) E λ , n + 1 ( x ) x m d x = 0 , 0 m < n + 1 , 1 1 w λ ( x ) P λ , n ( x ) E λ , n + 1 ( x ) x m d x 0 , m = n + 1 . In this paper, we investigate asymptotic properties of derivatives of the Stieltjes polynomials E λ , n + 1 ( x ) and the product E λ , n + 1 ( x ) P λ , n ( x ) . Especially, we estimate the even-order derivative values of E λ , n + 1 ( x ) and E λ , n + 1 ( x ) P λ , n ( x ) at the zeros of E λ , n + 1 ( x ) and the product E λ , n + 1 ( x ) P λ , n ( x ) , respectively. Moreover, we estimate asymptotic representations for the odd derivatives values of E λ , n + 1 ( x ) and E λ , n + 1 ( x ) P λ , n ( x ) at the zeros of E λ , n + 1 ( x ) and E λ , n + 1 ( x ) P λ , n ( x ) on a closed subset of ( 1 , 1 ) , respectively. These estimates will play important roles in investigating convergence and divergence of the higher-order Hermite-Fejér interpolation polynomials.

Article information

Source
J. Appl. Math., Volume 2012 (2012), Article ID 482935, 25 pages.

Dates
First available in Project Euclid: 14 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.jam/1355495242

Digital Object Identifier
doi:10.1155/2012/482935

Mathematical Reviews number (MathSciNet)
MR2959981

Zentralblatt MATH identifier
1252.41004

Citation

Jung, Hee Sun; Sakai, Ryozi. Asymptotic Properties of Derivatives of the Stieltjes Polynomials. J. Appl. Math. 2012 (2012), Article ID 482935, 25 pages. doi:10.1155/2012/482935. https://projecteuclid.org/euclid.jam/1355495242


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