Please use this identifier to cite or link to this item: http://hdl.handle.net/10016/5950

 Google™ Scholar. Others By: Geronimo, Jeffrey S. - Lubinsky, D. S. - Marcellán, Francisco
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 Title: Asymptotics for Sobolev orthogonal polynomials for exponential weights Author(s): Geronimo, Jeffrey S.Lubinsky, D. S.Marcellán, Francisco Publisher: Springer Issued date: Aug-2005 Citation: Constructive Approximation, 2005, vol. 22, n. 3, p. 309-346 URI: http://hdl.handle.net/10016/5950 ISSN: 0176-4276 (Print)1432-0940 (Online) DOI: 10.1007/s00365-004-0578-1 Description: 38 pages, no figures.-- MSC2000 codes: 42C05, 33C25.MR#: MR2164139 (2006c:41040)Zbl#: Zbl 1105.42016 Abstract: Let $\lambda >0,\alpha >1$, and let $W( x) =\exp ( -\vert x\vert \alpha })$, $x\in \mbox{\smallbf R}$. Let $\psi \in L_{\infty }(\mbox{\smallbf R})$ be positive on a set of positive measure. For $n\geq 1$, one may form Sobolev orthonormal polynomials $( q_{n})$, associated with the Sobolev inner product $( f,g) =\int_{\mbox{\scriptsize\bf R}}fg( \psi W) 2}+\lambda \int_{\mbox{\scriptsize\bf R}}f \prime }g \prime }W 2}.$ We establish strong asymptotics for the $( q_{n})$ in terms of the ordinary orthonormal polynomials $( p_{n})$ for the weight $W 2}$, on and off the real line. More generally, we establish a close asymptotic relationship between $( p_{n})$ and $( q_{n})$ for exponential weights $W=\exp ( -Q)$ on a real interval $I$, under mild conditions on $Q$. The method is new and will apply to many situations beyond that treated in this paper. Sponsor: The work by F. Marcellan has been supported by Dirección General de Investigación (Ministerio de Ciencia y Technología) of Spain under grant BFM 2003-06335-C03-07, as well as NATO Collaborative grant PST.CLG 979738. J. Geronimo and D. Lubinsky, respectively, acknowledge support by NSF grants DMS-0200219 and DMS-0400446. Review: PeerReviewed Publisher version: http://dx.doi.org/10.1007/s00365-004-0578-1 Keywords: Orthogonal polynomialsSobolev normsAsymptotics Rights: © Springer Appears in Collections: DM - GAMA - Artículos de Revistas

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