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

Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2011, Article ID 294134, 19 pages doi: Research Article Jacobi-Sobolev Orthogonal Polynomials: Asymptotics for N-Coherence of Measures Bujar Xh. Fejzullahu1 and Francisco Marcellan2 ´ 1 2 Faculty of Mathematics and Sciences, University of Prishtina, Mother Teresa 5, 10000 Prishtina, Kosovo Departamento de Matem´ ticas, Escuela Polit´cnica Superior, Universidad Carlos III de Madrid, a e Avenida de la Universidad, 30, 28911 Leganes, Spain Correspondence should be addressed to Francisco Marcell´ n, pacomarc@ a Received 24 November 2010; Accepted 7 March 2011 Academic Editor: Alexander I. Domoshnitsky Copyright q 2011 B. Xh. Fejzullahu and F. Marcell´ n. This. | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2011 Article ID 294134 19 pages doi 2011 294134 Research Article Jacobi-Sobolev Orthogonal Polynomials Asymptotics for N-Coherence of Measures Bujar Xh. Fejzullahu1 and Francisco Marcellan2 1 Faculty of Mathematics and Sciences University of Prishtina Mother Teresa 5 10000 Prishtina Kosovo 2 Departamento de Matemáticas Escuela Politécnica Superior Universidad Carlos III de Madrid Avenida de la Universidad 30 28911 Leganes Spain Correspondence should be addressed to Francisco Marcellan pacomarc@ Received 24 November 2010 Accepted 7 March 2011 Academic Editor Alexander I. Domoshnitsky Copyright 2011 B. Xh. Fejzullahu and F. Marcellan. 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. Let us introduce the Sobolev-type inner product f g f g u Àff 8 2 where A. 0 and f - 1 a fid- If - 1 a 1 P 1 M J 8 1 - -1f W8vw - x I1 x dx f S 2 - -1f x 8 I - x I1 x k 1 x - èk Nk 1 dx 2M1 ZNoMk if AềkìgAềk with a f -1 ịk 1 and Mk 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. 1. Introduction For a nontrivial probability measure ơ supported on -1 1 we define the linear space ư dơ of all measurable functions f on -1 1 such that f LPda TO where Wf llíP dơ _ ess sup If x I . -1 x 1 if 1 p TO if p TO. Let us now introduce the Sobolev-type spaces see . 1 Chapter 3 in a more general framework 2 Journal of Inequalities and Applications wN f I w. if t d Allf li ỂẳM á f r 1 p k 1i 0 w f Ilf WN. max Ilf h y Wk-w where A 0 and dpa p x 1 - x a 1 xýdx

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