A Favard type theorem for orthogonal polynomials on the unit circle from a three term recurrence formula

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dc.contributor.author Castillo Rodríguez, Kenier
dc.contributor.author Costa, Marisa de Souza
dc.contributor.author Sri Ranga, Alagacone
dc.contributor.author Veronese, Daniel Oliveira
dc.date.accessioned 2016-07-20T10:20:09Z
dc.date.available 2016-09-01T22:00:08Z
dc.date.issued 2014-08
dc.identifier.bibliographicCitation Journal of Approximation Theory, 2014, v. 184, pp. 146-162
dc.identifier.issn 0021-9045
dc.identifier.uri http://hdl.handle.net/10016/23390
dc.description.abstract The objective of this manuscript is to study directly the Favard type theorem associated with the three term recurrence formula Rn+1(z)=[(1+icn+1)z+(1−icn+1)]Rn(z)−4dn+1zRn−1(z),n≥1, Turn MathJax off with R0(z)=1 and R1(z)=(1+ic1)z+(1−ic1) , where {cn}∞n=1 is a real sequence and {dn}∞n=1 is a positive chain sequence. We establish that there exists a unique nontrivial probability measure μ on the unit circle for which {Rn(z)−2(1−mn)Rn−1(z)} gives the sequence of orthogonal polynomials. Here, {mn}∞n=0 is the minimal parameter sequence of the positive chain sequence {dn}∞n=1 . The element d1 of the chain sequence, which does not affect the polynomials Rn , has an influence in the derived probability measure μ and hence, in the associated orthogonal polynomials on the unit circle. To be precise, if {Mn}∞n=0 is the maximal parameter sequence of the chain sequence, then the measure μ is such that M0 is the size of its mass at z=1 . An example is also provided to completely illustrate the results obtained.
dc.description.sponsorship The works of all four authors have been supported by grants from CAPES, CNPq and FAPESP of Brazil. K. Castillo has also received support from Dirección General de Investigación, Ministerio de Economía y Competitividad of Spain (Grant MTM2012–36732–C03–01) for his research
dc.format.extent 18
dc.format.mimetype application/pdf
dc.language.iso eng
dc.publisher Elsevier
dc.rights © Elsevier 2014
dc.rights Atribución-NoComercial-SinDerivadas 3.0 España
dc.rights.uri http://creativecommons.org/licenses/by-nc-nd/3.0/es/
dc.subject.other Szegő polynomials
dc.subject.other Kernel polynomials
dc.subject.other Para-orthogonal polynomials
dc.subject.other Chain sequences
dc.subject.other Continued fractions
dc.title A Favard type theorem for orthogonal polynomials on the unit circle from a three term recurrence formula
dc.type article
dc.relation.publisherversion http://dx.doi.org/10.1016/j.jat.2014.05.007
dc.subject.eciencia Matemáticas
dc.identifier.doi 10.1016/j.jat.2014.05.007
dc.rights.accessRights openAccess
dc.relation.projectID Gobierno de España. MTM2012-36732-C03-01
dc.type.version acceptedVersion
dc.identifier.publicationfirstpage 146
dc.identifier.publicationlastpage 162
dc.identifier.publicationtitle Journal of approximation theory
dc.identifier.publicationvolume 184
dc.identifier.uxxi AR/0000018017
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