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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
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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  • 1. A. K. Common and J. H. McCabe, The symmetric strong moment problem, J. Comput. Appl. Math., 67 (1996), 327-341. MR 1390189 (97e:30003)
  • 2. C.F. Bracciali, J. H. McCabe and A. Sri Ranga, On a symmetric in strong distributions, J. Comput. Appl. Math., 105 (1999), 187-198. MR 1690586 (2000e:30074)
  • 3. L. Daruis, J. Hernández, and F. Marcellán, Spectral transformations for Hermitian Toeplitz matrices, J. Comput. Appl. Math., 202 (2007), 155-176. MR 2319946 (2008c:42025)
  • 4. D.K. Dimitrov, M.V. Mello, and F.R. Rafaeli, Monotonicity of zeros of Jacobi-Sobolev type orthogonal polynomials, Appl. Numer. Math., 60 (2010), 263-276. MR 2602677 (2011c:33016)
  • 5. E. Godoy and F. Marcellán, An analogue of the Christoffel formula for polynomial modification of a measure on the unit circle, Boll. Un. Mat. Ital., 5-A (1991), 1-12. MR 1101005 (92c:42022)
  • 6. E. Godoy and F. Marcellán, Orthogonal polynomials and rational modifications of measures, Canad. J. Math., 45 (1993), 930-943. MR 1239908 (95a:42031)
  • 7. W.B. Jones, O. Njåstad and W.J. Thron, Moment theory, orthogonal polynomials, quadrature, and continued fractions associated with the unit circle, Bull. London Math. Soc., 21 (1989), 113-152. MR 976057 (90e:42027)
  • 8. W.B. Jones, W.J. Thron and H. Waadeland, A strong Stieltjes moment problem, Trans. Amer. Math. Soc., 261 (1980), 503-528. MR 580900 (81j:30055)
  • 9. X. Li and F. Marcellán, Representation of orthogonal polynomials for modified measures, Comm. Anal. Theory of Cont. Fract., 7 (1999), 9-22.
  • 10. F. Marcellán and J. Hernández, Christoffel transforms and Hermitian linear functionals, Mediterr. J. Math., 2 (2005), 451-458. MR 2192525 (2006h:42044)
  • 11. F. Marcellán and J. Hernández, Geronimus spectral transforms and measures on the complex plane, J. Comput. Appl. Math., 219 (2008), 441-456. MR 2441238 (2009k:42053)
  • 12. F.R. Rafaeli, Zeros de Polinômios Ortogonais na Reta Real, Doctoral Thesis, UNICAMP, 2010.
  • 13. A. Sri Ranga, E.X.L. de Andrade and J.H. McCabe, Some consequences of symmetry in strong distributions, J. Math. Anal. Appl., 193 (1995), 158-168. MR 1338505 (97a:42022)
  • 14. A.P. da Silva and A. Sri Ranga, Polynomials generated by a three term recurrence relation: bounds for complex zeros, Linear Algebra Appl., 397 (2005), 299-324. MR 2116465 (2005m:33017)
  • 15. G. Szegő, Orthogonal polynomials, Amer. Math. Soc. Colloq. Publ., Vol. 23, Providence, RI, 1975. Fourth Edition. MR 0372517 (51:8724)
  • 16. L. Vinet and A. Zhedanov, Szegő polynomials on the real axis, Integr. Transforms and Spec. Functions, 8 (1999), 149-164. MR 1730600 (2000h:33010)

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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

Regina Litz Lamblém
Affiliation: Universidade Estadual de Mato Grosso do Sul - UEMS, Brazil

Fernando Rodrigo Rafaeli
Affiliation: Departamento de Matemática, Estatística e Computação, FCT, Universidade Estadual Paulista - UNESP, Brazil

Alagacone Sri Ranga
Affiliation: Departamento de Ciências de Computação e Estatística, IBILCE, Universidade Estadual Paulista - UNESP, Brazil

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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