# 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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