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Szegő and para-orthogonal polynomials on the real line: Zeros and canonical spectral transformations


Authors: Kenier Castillo, Regina Litz Lamblém, Fernando Rodrigo Rafaeli and Alagacone Sri Ranga
Journal: Math. Comp. 81 (2012), 2229-2249
MSC (2010): Primary 42C05, 30B70; Secondary 30E05
DOI: https://doi.org/10.1090/S0025-5718-2012-02593-2
Published electronically: April 2, 2012
MathSciNet review: 2945153
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Abstract: We study polynomials which satisfy the same recurrence relation as the Szegő polynomials, however, with the restriction that the (reflection) coefficients in the recurrence are larger than one in modulus. Para-orthogonal polynomials that follow from these Szegő polynomials are also considered. With positive values for the reflection coefficients, zeros of the Szegő polynomials, para-orthogonal polynomials and associated quadrature rules are also studied. Finally, again with positive values for the reflection coefficients, interlacing properties of the Szegő polynomials and polynomials arising from canonical spectral transformations are obtained.


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Additional Information

Kenier Castillo
Affiliation: Departamento de Ciências de Computação e Estatística, IBILCE, Universidade Estadual Paulista - UNESP, Brazil
Address at time of publication: Departamento de Matemáticas, Escuela Politécnica Superior, Universidad Carlos III, Leganés-Madrid, Spain
Email: kcastill@math.uc3m.es

Regina Litz Lamblém
Affiliation: Universidade Estadual de Mato Grosso do Sul - UEMS, Brazil
Email: lamblem@uems.br

Fernando Rodrigo Rafaeli
Affiliation: Departamento de Matemática, Estatística e Computação, FCT, Universidade Estadual Paulista - UNESP, Brazil
Email: rafaeli@fct.unesp.br

Alagacone Sri Ranga
Affiliation: Departamento de Ciências de Computação e Estatística, IBILCE, Universidade Estadual Paulista - UNESP, Brazil
Email: ranga@ibilce.unesp.br

DOI: https://doi.org/10.1090/S0025-5718-2012-02593-2
Keywords: Szegő polynomials, para-orthogonal polynomials, reflection coefficients, canonical spectral transformations
Received by editor(s): April 20, 2011
Received by editor(s) in revised form: June 12, 2011, and July 19, 2011
Published electronically: April 2, 2012
Additional Notes: This work has been done in the framework of a joint project of Dirección General de Investigación, Ministerio de Educación Ciencia of Spain and the Brazilian Science Foundation CAPES, Project CAPES/DGU 160/08.
The work of the first author was supported by Dirección General de Investigación, Ministerio de Ciencia e Innovación of Spain, grant MTM2009-12740-C03-01.
The work of the second, third and fourth authors was supported by FAPESP under grant 2009/13832-9.
The work of the fourth author was supported by CNPq.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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