# Asymptotics for Sobolev orthogonal polynomials for exponential weights

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 dc.contributor.author Geronimo, Jeffrey S. dc.contributor.author Lubinsky, D. S. dc.contributor.author Marcellán Español, Francisco José dc.date.accessioned 2009-12-04T11:00:07Z dc.date.available 2009-12-04T11:00:07Z dc.date.issued 2005-08 dc.identifier.bibliographicCitation Constructive Approximation, 2005, vol. 22, n. 3, p. 309-346 dc.identifier.issn 0176-4276 (Print) dc.identifier.issn 1432-0940 (Online) dc.identifier.uri http://hdl.handle.net/10016/5950 dc.description 38 pages, no figures.-- MSC2000 codes: 42C05, 33C25. dc.description MR#: MR2164139 (2006c:41040) dc.description Zbl#: Zbl 1105.42016 dc.description.abstract ^aLet $\lambda >0,\alpha >1$, 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. dc.description.sponsorship 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. dc.format.mimetype application/pdf dc.language.iso eng dc.publisher Springer dc.rights © Springer dc.subject.other Orthogonal polynomials dc.subject.other Sobolev norms dc.subject.other Asymptotics dc.title Asymptotics for Sobolev orthogonal polynomials for exponential weights dc.type article dc.type.review PeerReviewed dc.description.status Publicado dc.relation.publisherversion http://dx.doi.org/10.1007/s00365-004-0578-1 dc.subject.eciencia Matemáticas dc.identifier.doi 10.1007/s00365-004-0578-1 dc.rights.accessRights openAccess
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