Journal of Inequalities and Applications
Volume 2011 (2011), Article ID 294134, 19 pages
doi:10.1155/2011/294134
Research Article

Jacobi-Sobolev Orthogonal Polynomials: Asymptotics for N-Coherence of Measures

Bujar Xh. Fejzullahu1 and Francisco Marcellán2

1Faculty of Mathematics and Sciences, University of Prishtina, Mother Teresa 5, 10000 Prishtina, Kosovo
2Departamento de Matemáticas, Escuela Politécnica Superior, Universidad Carlos III de Madrid, Avenida de la Universidad, 30, 28911 Leganes, Spain

Received 24 November 2010; Accepted 7 March 2011

Academic Editor: Alexander I. Domoshnitsky

Copyright © 2011 Bujar Xh. Fejzullahu and Francisco Marcellán. 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 us introduce the Sobolev-type inner product ⟨ 𝑓 , 𝑔 ⟩ = ⟨ 𝑓 , 𝑔 ⟩ 1 + 𝜆 ⟨ 𝑓 , 𝑔 ⟩ 2 , where 𝜆 > 0 and ⟨ 𝑓 , 𝑔 ⟩ 1 = ∫ 1 − 1 𝑓 ( 𝑥 ) 𝑔 ( 𝑥 ) ( 1 − 𝑥 ) 𝛼 ( 1 + 𝑥 ) 𝛽 𝑑 𝑥 , ⟨ 𝑓 , 𝑔 ⟩ 2 = ∫ 1 − 1 𝑓 ( 𝑥 ) 𝑔 ( 𝑥 ) ( ( 1 − 𝑥 ) 𝛼 + 1 ( 1 + 𝑥 ) 𝛽 + 1 ) / ( ∏ 𝑀 𝑘 = 1 | 𝑥 − > 𝜉 𝑘 | 𝑁 𝑘 + 1 ) 𝑑 𝑥 + ∑ 𝑀 𝑘 = 1 ∑ 𝑁 𝑘 𝑖 = 0 𝑀 𝑘 , 𝑖 𝑓 ( 𝑖 ) ( 𝜉 𝑘 ) 𝑔 ( 𝑖 ) ( 𝜉 𝑘 ) , with 𝛼 , 𝛽 > − 1 , | 𝜉 𝑘 | > 1 , and 𝑀 𝑘 , 𝑖 > 0 , for all 𝑘 , 𝑖 . A Mehler-Heine-type formula and the inner strong asymptotics on ( − 1 , 1 ) as well as some estimates for the polynomials orthogonal with respect to the above Sobolev inner product are obtained. Necessary conditions for the norm convergence of Fourier expansions in terms of such Sobolev orthogonal polynomials are given.