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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